Optimal. Leaf size=207 \[ \frac {4 i \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 i \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f \sqrt {a+i a \sinh (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3319, 4182, 2279, 2391} \[ \frac {4 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{f \sqrt {a+i a \sinh (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3319
Rule 4182
Rubi steps
\begin {align*} \int \frac {x}{\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 \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 \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 (2 \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}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (2 \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}{f \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {4 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 )}{f \sqrt {a+i a \sinh (e+f x)}}-\frac {\left (4 \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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {\left (4 \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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {4 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 )}{f \sqrt {a+i a \sinh (e+f x)}}+\frac {4 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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 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 )}{f^2 \sqrt {a+i a \sinh (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 221, normalized size = 1.07 \[ \frac {\sqrt {2} \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left (-2 i \left (\text {Li}_2\left (-\sqrt [4]{-1} e^{-\frac {e}{2}-\frac {f x}{2}}\right )-\text {Li}_2\left (\sqrt [4]{-1} e^{-\frac {e}{2}-\frac {f x}{2}}\right )\right )-\frac {1}{2} (2 i e+2 i f x+\pi ) \left (\log \left (1-\sqrt [4]{-1} e^{-\frac {e}{2}-\frac {f x}{2}}\right )-\log \left (\sqrt [4]{-1} e^{-\frac {e}{2}-\frac {f x}{2}}+1\right )\right )-2 e \tan ^{-1}\left (\frac {\tanh \left (\frac {1}{4} (e+f x)\right )+i}{\sqrt {2}}\right )+i \pi \tan ^{-1}\left (\frac {\tanh \left (\frac {1}{4} (e+f x)\right )+i}{\sqrt {2}}\right )\right )}{f^2 \sqrt {a+i a \sinh (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}} x e^{\left (f x + e\right )}}{a e^{\left (f x + e\right )} - i \, a}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {a +i a \sinh \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {x}{\sqrt {i \, a \sinh \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {x}{\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {x}{\sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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