Optimal. Leaf size=689 \[ -\frac {3 i \text {Li}_3\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i \text {Li}_3\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x \text {Li}_2\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i x \text {Li}_2\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}} \]
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Rubi [A] time = 0.46, antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3319, 4186, 3768, 3770, 4182, 2531, 2282, 6589} \[ \frac {3 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,-e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3319
Rule 3768
Rule 3770
Rule 4182
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{(a+i a \sinh (c+d x))^{5/2}} \, dx &=\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int x^2 \text {csch}^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \text {csch}^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{16 a^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int \text {csch}^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{32 a^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \int \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{12 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \tanh ^{-1}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x \log \left (1-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int x \log \left (1+e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \tanh ^{-1}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \text {Li}_2\left (-e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \int \text {Li}_2\left (e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right ) \, dx}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \tanh ^{-1}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}-\frac {\left (3 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{-i \left (\frac {i c}{2}+\frac {\pi }{4}\right )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}\\ &=\frac {3 x}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {5 \tan ^{-1}\left (\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{3 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^2 \tanh ^{-1}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {3 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {x \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {\tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{6 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.29, size = 482, normalized size = 0.70 \[ \frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (9 c^2 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-9 c^2 \log \left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}+1\right )+18 c^2 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-9 d^2 x^2 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+9 d^2 x^2 \log \left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}+1\right )+36 d x \text {Li}_2\left (-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-36 d x \text {Li}_2\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-72 \text {Li}_3\left (-(-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )+72 \text {Li}_3\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )-160 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (c+d x)}\right )\right )+24 d^2 x^2 \sinh \left (\frac {1}{2} (c+d x)\right )+\left (9 i d^2 x^2+36 d x-8 i\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3+2 \left (9 d^2 x^2-8\right ) \sinh \left (\frac {1}{2} (c+d x)\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+4 d x (4+3 i d x) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{48 d^3 (a+i a \sinh (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ \frac {{\left (24 \, a^{3} d^{3} e^{\left (4 \, d x + 4 \, c\right )} - 96 i \, a^{3} d^{3} e^{\left (3 \, d x + 3 \, c\right )} - 144 \, a^{3} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 96 i \, a^{3} d^{3} e^{\left (d x + c\right )} + 24 \, a^{3} d^{3}\right )} {\rm integral}\left (\frac {{\left (-9 i \, d^{2} x^{2} + 80 i\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} e^{\left (d x + c\right )}}{48 \, {\left (a^{3} d^{2} e^{\left (d x + c\right )} - i \, a^{3} d^{2}\right )}}, x\right ) + {\left ({\left (-9 i \, d^{2} x^{2} - 36 i \, d x + 8 i\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (33 \, d^{2} x^{2} + 140 \, d x - 8\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-33 i \, d^{2} x^{2} + 140 i \, d x + 8 i\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (9 \, d^{2} x^{2} - 36 \, d x - 8\right )} e^{\left (d x + c\right )}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}}{24 \, a^{3} d^{3} e^{\left (4 \, d x + 4 \, c\right )} - 96 i \, a^{3} d^{3} e^{\left (3 \, d x + 3 \, c\right )} - 144 \, a^{3} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 96 i \, a^{3} d^{3} e^{\left (d x + c\right )} + 24 \, a^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (i a \left (\sinh {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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