Optimal. Leaf size=131 \[ -\frac {2 i \sqrt {1+a^2 x^2}}{a (1-i a x)^2 \sqrt {c+a^2 c x^2}}+\frac {4 i \sqrt {1+a^2 x^2}}{a (1-i a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 45}
\begin {gather*} \frac {4 i \sqrt {a^2 x^2+1}}{a (1-i a x) \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1}}{a (1-i a x)^2 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 5181
Rule 5184
Rubi steps
\begin {align*} \int \frac {e^{5 i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{5 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {(1+i a x)^2}{(1-i a x)^3} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \left (\frac {4}{(1-i a x)^3}-\frac {4}{(1-i a x)^2}+\frac {1}{1-i a x}\right ) \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 i \sqrt {1+a^2 x^2}}{a (1-i a x)^2 \sqrt {c+a^2 c x^2}}+\frac {4 i \sqrt {1+a^2 x^2}}{a (1-i a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 69, normalized size = 0.53 \begin {gather*} \frac {i \sqrt {1+a^2 x^2} \left (-2+4 i a x+(i+a x)^2 \log (i+a x)\right )}{a (i+a x)^2 \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 84, normalized size = 0.64
method | result | size |
risch | \(\frac {\sqrt {a^{2} x^{2}+1}\, \left (-4 x -\frac {2 i}{a}\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{2}}+\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x +i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\) | \(82\) |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (i \ln \left (a x +i\right ) a^{2} x^{2}-2 \ln \left (a x +i\right ) a x -i \ln \left (a x +i\right )-4 a x -2 i\right )}{\sqrt {a^{2} x^{2}+1}\, c a \left (a x +i\right )^{2}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 364 vs. \(2 (107) = 214\).
time = 4.25, size = 364, normalized size = 2.78 \begin {gather*} \frac {-4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} a x^{2} + {\left (i \, a^{4} c x^{4} - 2 \, a^{3} c x^{3} - 2 \, a c x - i \, c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) + {\left (-i \, a^{4} c x^{4} + 2 \, a^{3} c x^{3} + 2 \, a c x + i \, c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right )}{2 \, {\left (a^{4} c x^{4} + 2 i \, a^{3} c x^{3} + 2 i \, a c x - c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i \left (\int \left (- \frac {i}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {5 a x}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {10 a^{3} x^{3}}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{5} x^{5}}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {10 i a^{2} x^{2}}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {5 i a^{4} x^{4}}{a^{4} x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 2 a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^5}{\sqrt {c\,a^2\,x^2+c}\,{\left (a^2\,x^2+1\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________