Optimal. Leaf size=106 \[ -\frac{4 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac{4 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3}{3 a f} \]
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Rubi [A] time = 0.184023, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {5559, 2190, 2531, 2282, 6589} \[ -\frac{4 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac{4 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}+\frac{i (e+f x)^3}{3 a f} \]
Antiderivative was successfully verified.
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Rule 5559
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=\frac{i (e+f x)^3}{3 a f}+2 \int \frac{e^{c+d x} (e+f x)^2}{a+i a e^{c+d x}} \, dx\\ &=\frac{i (e+f x)^3}{3 a f}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}+\frac{(4 i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a d}\\ &=\frac{i (e+f x)^3}{3 a f}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}-\frac{4 i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^2}+\frac{\left (4 i f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{i (e+f x)^3}{3 a f}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}-\frac{4 i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^2}+\frac{\left (4 i f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac{i (e+f x)^3}{3 a f}-\frac{2 i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d}-\frac{4 i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{a d^2}+\frac{4 i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{a d^3}\\ \end{align*}
Mathematica [A] time = 0.0498098, size = 94, normalized size = 0.89 \[ \frac{i \left (-12 d f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )+12 f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )+d^2 (e+f x)^2 \left (d (e+f x)-6 f \log \left (1+i e^{c+d x}\right )\right )\right )}{3 a d^3 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 393, normalized size = 3.7 \begin{align*}{\frac{-2\,i{f}^{2}\ln \left ( 1+i{{\rm e}^{dx+c}} \right ){x}^{2}}{da}}+{\frac{2\,ief{c}^{2}}{a{d}^{2}}}-{\frac{4\,i{f}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{dx+c}} \right ) x}{a{d}^{2}}}-{\frac{2\,i\ln \left ({{\rm e}^{dx+c}}-i \right ){e}^{2}}{da}}-{\frac{2\,i{f}^{2}{c}^{2}x}{a{d}^{2}}}+{\frac{4\,i{f}^{2}{\it polylog} \left ( 3,-i{{\rm e}^{dx+c}} \right ) }{a{d}^{3}}}-{\frac{2\,i{f}^{2}{c}^{2}\ln \left ({{\rm e}^{dx+c}}-i \right ) }{a{d}^{3}}}+{\frac{2\,i\ln \left ({{\rm e}^{dx+c}} \right ){e}^{2}}{da}}-{\frac{i{e}^{2}x}{a}}-{\frac{4\,iefc\ln \left ({{\rm e}^{dx+c}} \right ) }{a{d}^{2}}}-{\frac{4\,ief\ln \left ( 1+i{{\rm e}^{dx+c}} \right ) x}{da}}+{\frac{{\frac{i}{3}}{x}^{3}{f}^{2}}{a}}-{\frac{4\,ief\ln \left ( 1+i{{\rm e}^{dx+c}} \right ) c}{a{d}^{2}}}-{\frac{4\,ief{\it polylog} \left ( 2,-i{{\rm e}^{dx+c}} \right ) }{a{d}^{2}}}+{\frac{2\,i{f}^{2}{c}^{2}\ln \left ({{\rm e}^{dx+c}} \right ) }{a{d}^{3}}}-{\frac{{\frac{4\,i}{3}}{f}^{2}{c}^{3}}{a{d}^{3}}}+{\frac{4\,iefc\ln \left ({{\rm e}^{dx+c}}-i \right ) }{a{d}^{2}}}+{\frac{ief{x}^{2}}{a}}+{\frac{4\,iefcx}{da}}+{\frac{2\,i{f}^{2}{c}^{2}\ln \left ( 1+i{{\rm e}^{dx+c}} \right ) }{a{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56212, size = 221, normalized size = 2.08 \begin{align*} -\frac{i \, e^{2} \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} - \frac{i \, f^{2} x^{3} + 3 i \, e f x^{2}}{3 \, a} - \frac{4 i \,{\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) +{\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} - \frac{2 i \,{\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x{\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \,{\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac{2 i \, d^{3} f^{2} x^{3} + 6 i \, d^{3} e f x^{2}}{3 \, a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.22861, size = 482, normalized size = 4.55 \begin{align*} \frac{i \, d^{3} f^{2} x^{3} + 3 i \, d^{3} e f x^{2} + 3 i \, d^{3} e^{2} x + 6 i \, c d^{2} e^{2} - 6 i \, c^{2} d e f + 2 i \, c^{3} f^{2} + 12 i \, f^{2}{\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right ) +{\left (-12 i \, d f^{2} x - 12 i \, d e f\right )}{\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) +{\left (-6 i \, d^{2} e^{2} + 12 i \, c d e f - 6 i \, c^{2} f^{2}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) +{\left (-6 i \, d^{2} f^{2} x^{2} - 12 i \, d^{2} e f x - 12 i \, c d e f + 6 i \, c^{2} f^{2}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{3 \, a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{e^{2} \cosh{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx + \int \frac{f^{2} x^{2} \cosh{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx + \int \frac{2 e f x \cosh{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx}{a} \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 (f x + e\right )}^{2} \cosh \left (d x + c\right )}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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