Optimal. Leaf size=122 \[ \frac{e^{d+e x} F^{c (a+b x)} (e-b c \log (F)) \, _2F_1\left (1,\frac{e+b c \log (F)}{2 e};\frac{1}{2} \left (\frac{b c \log (F)}{e}+3\right );e^{2 (d+e x)}\right )}{e^2}-\frac{b c \log (F) \text{csch}(d+e x) F^{c (a+b x)}}{2 e^2}-\frac{\coth (d+e x) \text{csch}(d+e x) F^{c (a+b x)}}{2 e} \]
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Rubi [A] time = 0.0530388, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {5491, 5493} \[ \frac{e^{d+e x} F^{c (a+b x)} (e-b c \log (F)) \, _2F_1\left (1,\frac{e+b c \log (F)}{2 e};\frac{1}{2} \left (\frac{b c \log (F)}{e}+3\right );e^{2 (d+e x)}\right )}{e^2}-\frac{b c \log (F) \text{csch}(d+e x) F^{c (a+b x)}}{2 e^2}-\frac{\coth (d+e x) \text{csch}(d+e x) F^{c (a+b x)}}{2 e} \]
Antiderivative was successfully verified.
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Rule 5491
Rule 5493
Rubi steps
\begin{align*} \int F^{c (a+b x)} \text{csch}^3(d+e x) \, dx &=-\frac{F^{c (a+b x)} \coth (d+e x) \text{csch}(d+e x)}{2 e}-\frac{b c F^{c (a+b x)} \text{csch}(d+e x) \log (F)}{2 e^2}-\frac{1}{2} \left (1-\frac{b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \text{csch}(d+e x) \, dx\\ &=-\frac{F^{c (a+b x)} \coth (d+e x) \text{csch}(d+e x)}{2 e}-\frac{b c F^{c (a+b x)} \text{csch}(d+e x) \log (F)}{2 e^2}+\frac{e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac{e+b c \log (F)}{2 e};\frac{1}{2} \left (3+\frac{b c \log (F)}{e}\right );e^{2 (d+e x)}\right ) (e-b c \log (F))}{e^2}\\ \end{align*}
Mathematica [B] time = 19.6679, size = 299, normalized size = 2.45 \[ \frac{F^{c (a+b x)} \left (\frac{4 \left (e^2-b^2 c^2 \log ^2(F)\right ) \left ((\sinh (d)+\cosh (d)-1) \, _2F_1\left (1,\frac{b c \log (F)}{e};\frac{b c \log (F)}{e}+1;\cosh (d+e x)+\sinh (d+e x)\right )+1\right )}{b c \log (F) (\sinh (d)+\cosh (d)-1)}+\frac{4 \left (e^2-b^2 c^2 \log ^2(F)\right ) \left (1-(\sinh (d)+\cosh (d)+1) \, _2F_1\left (1,\frac{b c \log (F)}{e};\frac{b c \log (F)}{e}+1;-\cosh (d+e x)-\sinh (d+e x)\right )\right )}{b c \log (F) (\sinh (d)+\cosh (d)+1)}+\text{csch}(d) \left (4 b c \log (F)-\frac{4 e^2}{b c \log (F)}\right )+2 b c \text{sech}\left (\frac{d}{2}\right ) \log (F) \sinh \left (\frac{e x}{2}\right ) \text{sech}\left (\frac{1}{2} (d+e x)\right )+2 b c \text{csch}\left (\frac{d}{2}\right ) \log (F) \sinh \left (\frac{e x}{2}\right ) \text{csch}\left (\frac{1}{2} (d+e x)\right )-4 b c \text{csch}(d) \log (F)-e \text{csch}^2\left (\frac{1}{2} (d+e x)\right )-e \text{sech}^2\left (\frac{1}{2} (d+e x)\right )\right )}{8 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ({\rm csch} \left (ex+d\right ) \right ) ^{3}\, dx \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*} 48 \,{\left (F^{a c} b c e e^{d} \log \left (F\right ) + F^{a c} e^{2} e^{d}\right )} \int \frac{e^{\left (b c x \log \left (F\right ) + e x\right )}}{b^{2} c^{2} \log \left (F\right )^{2} - 8 \, b c e \log \left (F\right ) + 15 \, e^{2} +{\left (b^{2} c^{2} e^{\left (8 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (8 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (8 \, d\right )}\right )} e^{\left (8 \, e x\right )} - 4 \,{\left (b^{2} c^{2} e^{\left (6 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (6 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 6 \,{\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (4 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} - 4 \,{\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (2 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}}\,{d x} - \frac{8 \,{\left (6 \, F^{a c} e e^{\left (e x + d\right )} +{\left (F^{a c} b c e^{\left (3 \, d\right )} \log \left (F\right ) - 5 \, F^{a c} e e^{\left (3 \, d\right )}\right )} e^{\left (3 \, e x\right )}\right )} F^{b c x}}{b^{2} c^{2} \log \left (F\right )^{2} - 8 \, b c e \log \left (F\right ) + 15 \, e^{2} -{\left (b^{2} c^{2} e^{\left (6 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (6 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \,{\left (b^{2} c^{2} e^{\left (4 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (4 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} - 3 \,{\left (b^{2} c^{2} e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 8 \, b c e e^{\left (2 \, d\right )} \log \left (F\right ) + 15 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \operatorname{csch}\left (e x + d\right )^{3}, x\right ) \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*} \int F^{c \left (a + b x\right )} \operatorname{csch}^{3}{\left (d + e x \right )}\, dx \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 F^{{\left (b x + a\right )} c} \operatorname{csch}\left (e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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