Optimal. Leaf size=506 \[ -\frac {4 i \text {Li}_3\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i \text {Li}_3\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {2 i x \text {Li}_2\left (-e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {2 i x \text {Li}_2\left (e^{\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}}\right ) \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f \sqrt {a+i a \sinh (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3319, 4186, 3770, 4182, 2531, 2282, 6589} \[ \frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,-e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (2,e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,-e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {4 i \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {PolyLog}\left (3,e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 a f \sqrt {a+i a \sinh (e+f x)}}+\frac {i x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {f x}{2}+\frac {1}{4} (2 e-i \pi )}\right )}{a f \sqrt {a+i a \sinh (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 3319
Rule 3770
Rule 4182
Rule 4186
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{(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^2 \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 {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}+\frac {x^2 \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^2 \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 (2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \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 {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\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^2 \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^2 \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 \log \left (1-e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{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 \log \left (1+e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right ) \, dx}{a f \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\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^2 \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 {2 i x \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 {2 i x \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^2 \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 (2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \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 (2 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \int \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 {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\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^2 \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 {2 i x \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 {2 i x \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^2 \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 (4 \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^3 \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 {\text {Li}_2(x)}{x} \, dx,x,e^{-i \left (\frac {i e}{2}+\frac {\pi }{4}\right )+\frac {f x}{2}}\right )}{a f^3 \sqrt {a+i a \sinh (e+f x)}}\\ &=\frac {2 x}{a f^2 \sqrt {a+i a \sinh (e+f x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\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^2 \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 {2 i x \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 {2 i x \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 {4 i \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 {4 i \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^2 \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*}
________________________________________________________________________________________
Mathematica [A] time = 1.68, size = 384, normalized size = 0.76 \[ \frac {\left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\left (\frac {1}{2}-\frac {i}{2}\right ) (-1)^{3/4} \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (e^2 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-e^2 \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )+2 e^2 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-f^2 x^2 \log \left (1-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+f^2 x^2 \log \left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}+1\right )+4 f x \text {Li}_2\left (-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-4 f x \text {Li}_2\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-8 \text {Li}_3\left (-(-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )+8 \text {Li}_3\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )-16 \tanh ^{-1}\left ((-1)^{3/4} e^{\frac {1}{2} (e+f x)}\right )\right )+2 f^2 x^2 \sinh \left (\frac {1}{2} (e+f x)\right )+f x (4+i f x) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{2 f^3 (a+i a \sinh (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} e^{\left (f x + e\right )} - a^{2} f^{2}\right )} {\rm integral}\left (\frac {{\left (-i \, f^{2} x^{2} + 8 i\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}} e^{\left (f x + e\right )}}{2 \, {\left (a^{2} f^{2} e^{\left (f x + e\right )} - i \, a^{2} f^{2}\right )}}, x\right ) + {\left ({\left (-i \, f x^{2} - 4 i \, x\right )} e^{\left (2 \, f x + 2 \, e\right )} + {\left (f x^{2} - 4 \, x\right )} e^{\left (f x + e\right )}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-f x - e\right )}}}{a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 2 i \, a^{2} f^{2} e^{\left (f x + e\right )} - a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________