Optimal. Leaf size=807 \[ \frac {\tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{2 a f \sqrt {i \sinh (e+f x) a+a}}+\frac {i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{a f \sqrt {i \sinh (e+f x) a+a}}+\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}+\frac {3 x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}+\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_4\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {Li}_4\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}} \]
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Rubi [A] time = 0.44, antiderivative size = 807, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3319, 4186, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac {\tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{2 a f \sqrt {i \sinh (e+f x) a+a}}+\frac {i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x^3}{a f \sqrt {i \sinh (e+f x) a+a}}+\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {3 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}+\frac {3 x^2}{a f^2 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}+\frac {12 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) x}{a f^3 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (4,-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (4,e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {i \sinh (e+f x) a+a}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3319
Rule 4182
Rule 4186
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{(a+i a \sinh (e+f x))^{3/2}} \, dx &=-\frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \int x^3 \text {csch}^3\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{2 a \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \int x^3 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{4 a \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (3 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x^2 \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (3 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x^2 \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{2 a f \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {Li}_2\left (-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (6 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int x \text {Li}_2\left (e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^2 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \text {Li}_3\left (-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (12 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \text {Li}_3\left (e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f^3 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (24 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {3 x^2}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i x \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^3 \tanh ^{-1}\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {3 i x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {12 i x \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_4\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}-\frac {24 i \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \text {Li}_4\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right )}{a f^4 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.15, size = 546, normalized size = 0.68 \[ \frac {\left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (-1)^{3/4} \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (e^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-e^3 \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )+2 e^3 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-f^3 x^3 \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )-6 \left (f^2 x^2-8\right ) \text {Li}_2\left (-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+6 \left (f^2 x^2-8\right ) \text {Li}_2\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \text {Li}_3\left (-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 f x \text {Li}_3\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-48 \text {Li}_4\left (-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+48 \text {Li}_4\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-24 e \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 e \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )-24 f x \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+24 f x \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )-48 e \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right )+2 f^3 x^3 \sinh \left (\frac {1}{2} (e+f x)\right )+f^2 x^2 (6+i f x) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{2 f^4 (a+i a \sinh (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} e^{\left (f x + e\right )} - a^{2} f^{2}\right )} {\rm integral}\left (\frac {{\left (-i \, f^{2} x^{3} + 24 i \, x\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}} e^{\left (f x + e\right )}}{2 \, {\left (a^{2} f^{2} e^{\left (f x + e\right )} - i \, a^{2} f^{2}\right )}}, x\right ) + {\left ({\left (-i \, f x^{3} - 6 i \, x^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (f x^{3} - 6 \, x^{2}\right )} e^{\left (f x + e\right )}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}}}{a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} e^{\left (f x + e\right )} - a^{2} f^{2}} \]
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 \[ \int \frac {x^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{3}}{\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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