Implement algorithm="fricas" for integration

src/sage/symbolic/integration/external.py

@@ -68,3 +68,47 @@ def mma_free_integrator(expression, v, a=None, b=None):
         return ans
     except TypeError:
         raise ValueError("Unable to parse: %s" % mexpr)
+
+
+def fricas_integrator(expression, v, a=None, b=None):
+    """
+    Integration using FriCAS
+
+    EXAMPLES::
+
+        sage: from sage.symbolic.integration.external import fricas_integrator  # optional - fricas
+        sage: fricas_integrator(sin(x), x)                                      # optional - fricas
+        -cos(x)
+        sage: fricas_integrator(cos(x), x)                                      # optional - fricas
+        sin(x)
+        sage: fricas_integrator(1/(x^2-2), x, 0, 1)                             # optional - fricas
+        1/4*(log(3*sqrt(2) - 4) - log(sqrt(2)))*sqrt(2)
+        sage: fricas_integrator(1/(x^2+6), x, -oo, oo)                          # optional - fricas
+        1/6*pi*sqrt(6)
+    """
+    if not isinstance(expression, Expression):
+        expression = SR(expression)
+    if a is None:
+        result = expression._fricas_().integrate(v)
+    else:
+        import sage.rings.infinity
+        if a == sage.rings.infinity.PlusInfinity():
+            a = "%plusInfinity"
+        elif a == sage.rings.infinity.MinusInfinity():
+            a = "%minusInfinity"
+        if b == sage.rings.infinity.PlusInfinity():
+            b = "%plusInfinity"
+        elif b == sage.rings.infinity.MinusInfinity():
+            b = "%minusInfinity"
+
+        result = expression._fricas_().integrate("{}={}..{}".format(v, a, b))
+    locals = {str(v): v for v in expression.variables()}
+    if str(result) == "potentialPole":
+        raise ValueError("The integrand has a potential pole"
+                         " in the integration interval")
+    parsed_result = result.unparsed_input_form()
+    import sage.misc.sage_eval
+    try:
+        return sage.misc.sage_eval.sage_eval(parsed_result, locals=locals)
+    except:
+        raise ValueError("Unable to parse: {}".format(parsed_result))

src/sage/symbolic/integration/integral.py

@@ -27,6 +27,7 @@
 available_integrators['maxima'] = external.maxima_integrator
 available_integrators['sympy'] = external.sympy_integrator
 available_integrators['mathematica_free'] = external.mma_free_integrator
+available_integrators['fricas'] = external.fricas_integrator
 
 ######################################################
 #
@@ -344,6 +345,8 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
 
        - 'mathematica_free' - use http://integrals.wolfram.com/
 
+       - 'fricas' - use FriCAS (the optional fricas spkg has to be installed) 
+
     To prevent automatic evaluation use the ``hold`` argument.
 
      EXAMPLES::
@@ -518,6 +521,28 @@ def integrate(expression, v=None, a=None, b=None, algorithm=None, hold=False):
         sage: integral(e^(-x^2),(x, 0, 0.1))
         0.05623145800914245*sqrt(pi)
 
+    An example of an integral that fricas can integrate, but the
+    default integrator cannot::
+
+        sage: f(x) = sqrt(x+sqrt(1+x^2))/x
+        sage: integrate(f(x), x, algorithm="fricas")      # optional - fricas
+        2*sqrt(x + sqrt(x^2 + 1)) + log(sqrt(x + sqrt(x^2 + 1)) - 1)
+        - log(sqrt(x + sqrt(x^2 + 1)) + 1) - 2*arctan(sqrt(x + sqrt(x^2 + 1)))
+
+    The following definite integral is not found with the
+    default integrator::
+
+        sage: f(x) = (x^4 - 3*x^2 + 6) / (x^6 - 5*x^4 + 5*x^2 + 4)
+        sage: integrate(f(x), x, 1, 2)
+        integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)
+
+    Both fricas and sympy give the correct result::
+
+        sage: integrate(f(x), x, 1, 2, algorithm="fricas")  # optional - fricas
+        -1/2*pi + arctan(1/2) + arctan(2) + arctan(5) + arctan(8)
+        sage: integrate(f(x), x, 1, 2, algorithm="sympy")
+        -1/2*pi + arctan(1/2) + arctan(2) + arctan(5) + arctan(8)
+
     ALIASES: integral() and integrate() are the same.
 
     EXAMPLES: