Optimal. Leaf size=241 \[ -\frac{4 d^2 \text{PolyLog}\left (2,-i e^{e+f x}\right )}{3 a^2 f^3}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f}+\frac{(c+d x)^2}{3 a^2 f}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^3} \]
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Rubi [A] time = 0.277598, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3318, 4186, 3767, 8, 4184, 3716, 2190, 2279, 2391} \[ -\frac{4 d^2 \text{PolyLog}\left (2,-i e^{e+f x}\right )}{3 a^2 f^3}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f}+\frac{(c+d x)^2}{3 a^2 f}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^3} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 4186
Rule 3767
Rule 8
Rule 4184
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx &=\frac{\int (c+d x)^2 \csc ^4\left (\frac{1}{2} \left (i e+\frac{\pi }{2}\right )+\frac{i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\int (c+d x)^2 \text{csch}^2\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{6 a^2}+\frac{d^2 \int \text{csch}^2\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{3 a^2 f^2}\\ &=\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\left (2 i d^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \coth \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right )\right )}{3 a^2 f^3}-\frac{(2 d) \int (c+d x) \coth \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac{(c+d x)^2}{3 a^2 f}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{(4 i d) \int \frac{e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1+i e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=\frac{(c+d x)^2}{3 a^2 f}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^2\right ) \int \log \left (1+i e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=\frac{(c+d x)^2}{3 a^2 f}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=\frac{(c+d x)^2}{3 a^2 f}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}-\frac{4 d^2 \text{Li}_2\left (-i e^{e+f x}\right )}{3 a^2 f^3}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f^3}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{6 a^2 f}\\ \end{align*}
Mathematica [A] time = 3.55659, size = 269, normalized size = 1.12 \[ \frac{4 d^2 \text{PolyLog}\left (2,i e^{-e-f x}\right )+\frac{i \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cosh \left (e+\frac{3 f x}{2}\right )+\sinh \left (\frac{f x}{2}\right ) \left (3 c^2 f^2+6 c d f^2 x+d^2 \left (3 f^2 x^2-4\right )\right )+2 i d f (c+d x) \sinh \left (e+\frac{f x}{2}\right )+2 d f (c+d x) \cosh \left (\frac{f x}{2}\right )+2 i d^2 \cosh \left (e+\frac{f x}{2}\right )}{\left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right ) \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{2 i f (c+d x) \left (f (c+d x)+2 d \left (1+i e^e\right ) \log \left (1-i e^{-e-f x}\right )\right )}{e^e-i}}{3 a^2 f^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 374, normalized size = 1.6 \begin{align*}{\frac{-2\,i{f}^{2}{d}^{2}{x}^{2}-4\,f{d}^{2}x{{\rm e}^{fx+e}}-4\,fcd{{\rm e}^{fx+e}}-8\,{d}^{2}{{\rm e}^{fx+e}}+4\,i{d}^{2}-4\,i{d}^{2}{{\rm e}^{2\,fx+2\,e}}-2\,i{f}^{2}{c}^{2}-4\,i{f}^{2}cdx-4\,if{d}^{2}x{{\rm e}^{2\,fx+2\,e}}-4\,ifcd{{\rm e}^{2\,fx+2\,e}}+6\,{f}^{2}{d}^{2}{x}^{2}{{\rm e}^{fx+e}}+12\,{f}^{2}cdx{{\rm e}^{fx+e}}+6\,{f}^{2}{c}^{2}{{\rm e}^{fx+e}}}{3\, \left ({{\rm e}^{fx+e}}-i \right ) ^{3}{f}^{3}{a}^{2}}}-{\frac{4\,d\ln \left ({{\rm e}^{fx+e}}-i \right ) c}{3\,{a}^{2}{f}^{2}}}+{\frac{4\,d\ln \left ({{\rm e}^{fx+e}} \right ) c}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{x}^{2}}{3\,f{a}^{2}}}+{\frac{4\,{d}^{2}ex}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{e}^{2}}{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) x}{3\,{a}^{2}{f}^{2}}}-{\frac{4\,{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) e}{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}}+{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{fx+e}}-i \right ) }{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{2}{\left (\frac{-2 i \, f^{2} x^{2} -{\left (4 i \, f x e^{\left (2 \, e\right )} + 4 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 2 \,{\left (3 \, f^{2} x^{2} e^{e} - 2 \, f x e^{e} - 4 \, e^{e}\right )} e^{\left (f x\right )} + 4 i}{3 \, a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{3} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{3}} - 4 i \, \int \frac{x}{3 \,{\left (a^{2} f e^{\left (f x + e\right )} - i \, a^{2} f\right )}}\,{d x}\right )} + \frac{2}{3} \, c d{\left (\frac{3 \,{\left (2 \, f x e^{\left (3 \, f x + 3 \, e\right )} +{\left (-6 i \, f x e^{\left (2 \, e\right )} - 2 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 2 \, e^{\left (f x + e\right )}\right )}}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} - \frac{2 \, \log \left (-i \,{\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + c^{2}{\left (\frac{6 \, e^{\left (-f x - e\right )}}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f} + \frac{2 i}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.50375, size = 1162, normalized size = 4.82 \begin{align*} \frac{-2 i \, d^{2} e^{2} + 4 i \, c d e f - 2 i \, c^{2} f^{2} + 4 i \, d^{2} -{\left (4 \, d^{2} e^{\left (3 \, f x + 3 \, e\right )} - 12 i \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 12 \, d^{2} e^{\left (f x + e\right )} + 4 i \, d^{2}\right )}{\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) + 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-6 i \, d^{2} f^{2} x^{2} + 6 i \, d^{2} e^{2} - 4 i \, d^{2} +{\left (-12 i \, c d e - 4 i \, c d\right )} f +{\left (-12 i \, c d f^{2} - 4 i \, d^{2} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (3 \, d^{2} e^{2} + 3 \, c^{2} f^{2} - 2 \, d^{2} f x - 4 \, d^{2} - 2 \,{\left (3 \, c d e + c d\right )} f\right )} e^{\left (f x + e\right )} +{\left (4 i \, d^{2} e - 4 i \, c d f + 4 \,{\left (d^{2} e - c d f\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-12 i \, d^{2} e + 12 i \, c d f\right )} e^{\left (2 \, f x + 2 \, e\right )} - 12 \,{\left (d^{2} e - c d f\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) +{\left (-4 i \, d^{2} f x - 4 i \, d^{2} e - 4 \,{\left (d^{2} f x + d^{2} e\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (12 i \, d^{2} f x + 12 i \, d^{2} e\right )} e^{\left (2 \, f x + 2 \, e\right )} + 12 \,{\left (d^{2} f x + d^{2} e\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right )}{3 \, a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{3} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{3}} \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 )}^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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