Optimal. Leaf size=132 \[ -\frac{12 d^2 (c+d x) \text{PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac{12 d^3 \text{PolyLog}\left (3,-i e^{e+f x}\right )}{a f^4}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{a f}+\frac{(c+d x)^3}{a f} \]
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Rubi [A] time = 0.302292, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3318, 4184, 3716, 2190, 2531, 2282, 6589} \[ -\frac{12 d^2 (c+d x) \text{PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac{12 d^3 \text{PolyLog}\left (3,-i e^{e+f x}\right )}{a f^4}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{a f}+\frac{(c+d x)^3}{a f} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 4184
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{a+i a \sinh (e+f x)} \, dx &=\frac{\int (c+d x)^3 \csc ^2\left (\frac{1}{2} \left (i e+\frac{\pi }{2}\right )+\frac{i f x}{2}\right ) \, dx}{2 a}\\ &=\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}-\frac{(3 d) \int (c+d x)^2 \coth \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{a f}\\ &=\frac{(c+d x)^3}{a f}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}-\frac{(6 i d) \int \frac{e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)^2}{1+i e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a f}\\ &=\frac{(c+d x)^3}{a f}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{\left (12 d^2\right ) \int (c+d x) \log \left (1+i e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=\frac{(c+d x)^3}{a f}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac{12 d^2 (c+d x) \text{Li}_2\left (-i e^{e+f x}\right )}{a f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{\left (12 d^3\right ) \int \text{Li}_2\left (-i e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a f^3}\\ &=\frac{(c+d x)^3}{a f}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac{12 d^2 (c+d x) \text{Li}_2\left (-i e^{e+f x}\right )}{a f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{\left (12 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a f^4}\\ &=\frac{(c+d x)^3}{a f}-\frac{6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac{12 d^2 (c+d x) \text{Li}_2\left (-i e^{e+f x}\right )}{a f^3}+\frac{12 d^3 \text{Li}_3\left (-i e^{e+f x}\right )}{a f^4}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{a f}\\ \end{align*}
Mathematica [A] time = 2.89412, size = 206, normalized size = 1.56 \[ \frac{2 \left (\frac{3 d e^e \left (-\frac{2 i d e^{-e} \left (e^e-i\right ) \left (f (c+d x) \text{PolyLog}\left (2,i e^{-e-f x}\right )+d \text{PolyLog}\left (3,i e^{-e-f x}\right )\right )}{f^3}+\frac{\left (e^{-e}+i\right ) (c+d x)^2 \log \left (1-i e^{-e-f x}\right )}{f}+\frac{e^{-e} (c+d x)^3}{3 d}\right )}{-1-i e^e}+\frac{(c+d x)^3 \sinh \left (\frac{f x}{2}\right )}{\left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right ) \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )}\right )}{a f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 435, normalized size = 3.3 \begin{align*}{\frac{2\,i \left ({d}^{3}{x}^{3}+3\,c{d}^{2}{x}^{2}+3\,{c}^{2}dx+{c}^{3} \right ) }{fa \left ({{\rm e}^{fx+e}}-i \right ) }}-12\,{\frac{c{d}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{fx+e}} \right ) }{a{f}^{3}}}-4\,{\frac{{d}^{3}{e}^{3}}{{f}^{4}a}}+6\,{\frac{d\ln \left ({{\rm e}^{fx+e}} \right ){c}^{2}}{a{f}^{2}}}+6\,{\frac{{d}^{3}{e}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) }{{f}^{4}a}}-12\,{\frac{c{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) x}{a{f}^{2}}}-12\,{\frac{c{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) e}{a{f}^{3}}}+12\,{\frac{c{d}^{2}ex}{a{f}^{2}}}+2\,{\frac{{d}^{3}{x}^{3}}{fa}}-6\,{\frac{{d}^{3}{e}^{2}x}{a{f}^{3}}}-12\,{\frac{c{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{a{f}^{3}}}+6\,{\frac{c{d}^{2}{x}^{2}}{fa}}+6\,{\frac{c{d}^{2}{e}^{2}}{a{f}^{3}}}-6\,{\frac{{d}^{3}{e}^{2}\ln \left ({{\rm e}^{fx+e}}-i \right ) }{{f}^{4}a}}+12\,{\frac{c{d}^{2}e\ln \left ({{\rm e}^{fx+e}}-i \right ) }{a{f}^{3}}}-12\,{\frac{{d}^{3}{\it polylog} \left ( 2,-i{{\rm e}^{fx+e}} \right ) x}{a{f}^{3}}}+12\,{\frac{{d}^{3}{\it polylog} \left ( 3,-i{{\rm e}^{fx+e}} \right ) }{{f}^{4}a}}-6\,{\frac{{d}^{3}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ){x}^{2}}{a{f}^{2}}}-6\,{\frac{d\ln \left ({{\rm e}^{fx+e}}-i \right ){c}^{2}}{a{f}^{2}}}+6\,{\frac{{d}^{3}{e}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{{f}^{4}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.68317, size = 320, normalized size = 2.42 \begin{align*} 6 \, c^{2} d{\left (\frac{x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} - i \, a f} - \frac{\log \left ({\left (e^{\left (f x + e\right )} - i\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} - \frac{2 \, c^{3}}{{\left (i \, a e^{\left (-f x - e\right )} - a\right )} f} + \frac{2 i \, d^{3} x^{3} + 6 i \, c d^{2} x^{2}}{a f e^{\left (f x + e\right )} - i \, a f} - \frac{12 \,{\left (f x \log \left (i \, e^{\left (f x + e\right )} + 1\right ) +{\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a f^{3}} - \frac{6 \,{\left (f^{2} x^{2} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + 2 \, f x{\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - 2 \,{\rm Li}_{3}(-i \, e^{\left (f x + e\right )})\right )} d^{3}}{a f^{4}} + \frac{2 \,{\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{a f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.72744, size = 863, normalized size = 6.54 \begin{align*} \frac{-2 i \, d^{3} e^{3} + 6 i \, c d^{2} e^{2} f - 6 i \, c^{2} d e f^{2} + 2 i \, c^{3} f^{3} +{\left (12 i \, d^{3} f x + 12 i \, c d^{2} f - 12 \,{\left (d^{3} f x + c d^{2} f\right )} e^{\left (f x + e\right )}\right )}{\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) + 2 \,{\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} e^{\left (f x + e\right )} +{\left (6 i \, d^{3} e^{2} - 12 i \, c d^{2} e f + 6 i \, c^{2} d f^{2} - 6 \,{\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) +{\left (6 i \, d^{3} f^{2} x^{2} + 12 i \, c d^{2} f^{2} x - 6 i \, d^{3} e^{2} + 12 i \, c d^{2} e f - 6 \,{\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) +{\left (12 \, d^{3} e^{\left (f x + e\right )} - 12 i \, d^{3}\right )}{\rm polylog}\left (3, -i \, e^{\left (f x + e\right )}\right )}{a f^{4} e^{\left (f x + e\right )} - i \, a f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{3}}{i \, a \sinh \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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