Optimal. Leaf size=100 \[ \frac{(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{1}{2} \left (\frac{d}{b}+3\right );e^{2 (a+b x)}\right )}{b^2}-\frac{d e^{c+d x} \text{csch}(a+b x)}{2 b^2}-\frac{e^{c+d x} \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.0489449, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5491, 5493} \[ \frac{(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{1}{2} \left (\frac{d}{b}+3\right );e^{2 (a+b x)}\right )}{b^2}-\frac{d e^{c+d x} \text{csch}(a+b x)}{2 b^2}-\frac{e^{c+d x} \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5491
Rule 5493
Rubi steps
\begin{align*} \int e^{c+d x} \text{csch}^3(a+b x) \, dx &=-\frac{d e^{c+d x} \text{csch}(a+b x)}{2 b^2}-\frac{e^{c+d x} \coth (a+b x) \text{csch}(a+b x)}{2 b}-\frac{1}{2} \left (1-\frac{d^2}{b^2}\right ) \int e^{c+d x} \text{csch}(a+b x) \, dx\\ &=-\frac{d e^{c+d x} \text{csch}(a+b x)}{2 b^2}-\frac{e^{c+d x} \coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{1}{2} \left (3+\frac{d}{b}\right );e^{2 (a+b x)}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 2.354, size = 94, normalized size = 0.94 \[ \frac{e^c \left (\frac{2 \text{csch}(a) (b-d) e^{x (b+d)} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )}{\coth (a)-1}-e^{d x} \text{csch}(a+b x) (b \coth (a+b x)+d)\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ({\rm csch} \left (bx+a\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 (b^{2} e^{c} + b d e^{c}\right )} \int \frac{e^{\left (b x + d x + a\right )}}{15 \, b^{2} - 8 \, b d + d^{2} +{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{8 \,{\left ({\left (5 \, b e^{c} - d e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} - 6 \, b e^{\left (b x + a + c\right )}\right )} e^{\left (d x\right )}}{15 \, b^{2} - 8 \, b d + d^{2} -{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\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 (\operatorname{csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}, 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*} e^{c} \int e^{d x} \operatorname{csch}^{3}{\left (a + b 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 \operatorname{csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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