Optimal. Leaf size=493 \[ \frac{12 i x^2 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 i x^2 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{48 i x \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{48 i x \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{96 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{96 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{4 i x^3 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f \sqrt{a+i a \sinh (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.260732, antiderivative size = 493, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3319, 4182, 2531, 6609, 2282, 6589} \[ \frac{12 i x^2 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 i x^2 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{48 i x \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{48 i x \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{96 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{96 i \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{4 i x^3 \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{f x}{2}+\frac{1}{4} (2 e-i \pi )}\right )}{f \sqrt{a+i a \sinh (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3319
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{a+i a \sinh (e+f x)}} \, dx &=\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}{\sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{4 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 )}{f \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^2 \log \left (1-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{f \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^2 \log \left (1+e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{f \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{4 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 )}{f \sqrt{a+i a \sinh (e+f x)}}+\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 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 )}{f^2 \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 ) \int x \text{Li}_2\left (-e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{f^2 \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 ) \int x \text{Li}_2\left (e^{-i \left (\frac{i e}{2}+\frac{\pi }{4}\right )+\frac{f x}{2}}\right ) \, dx}{f^2 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{4 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 )}{f \sqrt{a+i a \sinh (e+f x)}}+\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (48 \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}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (48 \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}{f^3 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{4 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 )}{f \sqrt{a+i a \sinh (e+f x)}}+\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}-\frac{\left (96 \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 )}{f^4 \sqrt{a+i a \sinh (e+f x)}}+\frac{\left (96 \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 )}{f^4 \sqrt{a+i a \sinh (e+f x)}}\\ &=\frac{4 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 )}{f \sqrt{a+i a \sinh (e+f x)}}+\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{12 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 )}{f^2 \sqrt{a+i a \sinh (e+f x)}}-\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{48 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 )}{f^3 \sqrt{a+i a \sinh (e+f x)}}+\frac{96 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 )}{f^4 \sqrt{a+i a \sinh (e+f x)}}-\frac{96 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 )}{f^4 \sqrt{a+i a \sinh (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.27041, size = 331, normalized size = 0.67 \[ \frac{(1-i) (-1)^{3/4} \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right ) \left (-6 f^2 x^2 \text{PolyLog}\left (2,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+6 f^2 x^2 \text{PolyLog}\left (2,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+24 f x \text{PolyLog}\left (3,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-24 f x \text{PolyLog}\left (3,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-48 \text{PolyLog}\left (4,-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+48 \text{PolyLog}\left (4,(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )+e^3 \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-e^3 \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )+2 i e^3 \tan ^{-1}\left (\sqrt [4]{-1} e^{\frac{1}{2} (e+f x)}\right )+f^3 x^3 \log \left (1-(-1)^{3/4} e^{\frac{1}{2} (e+f x)}\right )-f^3 x^3 \log \left ((-1)^{3/4} e^{\frac{1}{2} (e+f x)}+1\right )\right )}{f^4 \sqrt{a+i a \sinh (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3}{\frac{1}{\sqrt{a+ia\sinh \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{i \, a \sinh \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{2 i \, \sqrt{\frac{1}{2}} \sqrt{i \, a e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - i \, a} x^{3} e^{\left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{a e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a e^{\left (f x + e\right )} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a \left (i \sinh{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{i \, a \sinh \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]