Optimal. Leaf size=151 \[ -\frac{2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac{8 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac{8 e^{a+x (b+d)+c} \, _2F_1\left (3,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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Rubi [A] time = 0.343, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5511, 2251} \[ -\frac{2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac{8 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac{8 e^{a+x (b+d)+c} \, _2F_1\left (3,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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Rule 5511
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \coth ^2(a+b x) \text{csch}(a+b x) \, dx &=\int \left (\frac{8 e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3}+\frac{8 e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2}+\frac{2 e^{a+c+(b+d) x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=2 \int \frac{e^{a+c+(b+d) x}}{-1+e^{2 (a+b x)}} \, dx+8 \int \frac{e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3} \, dx+8 \int \frac{e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx\\ &=-\frac{2 e^{a+c+(b+d) x} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac{8 e^{a+c+(b+d) x} \, _2F_1\left (2,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac{8 e^{a+c+(b+d) x} \, _2F_1\left (3,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}\\ \end{align*}
Mathematica [A] time = 1.32094, size = 111, normalized size = 0.74 \[ -\frac{e^{c-\frac{a d}{b}} \left (2 \left (b^2+d^2\right ) e^{\frac{(b+d) (a+b x)}{b}} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )+(b+d) e^{d \left (\frac{a}{b}+x\right )} \text{csch}(a+b x) (b \coth (a+b x)+d)\right )}{2 b^2 (b+d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.136, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \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^{3} e^{c} + b d^{2} e^{c}\right )} \int \frac{e^{\left (b x + d x + a\right )}}{15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3} +{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \,{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \,{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \,{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{2 \,{\left ({\left (15 \, b^{2} e^{c} - 8 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \,{\left (10 \, b^{2} e^{c} + 3 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} +{\left (9 \, b^{2} e^{c} + 14 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (b x + a\right )}\right )} e^{\left (d x\right )}}{15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3} -{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\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 (\cosh \left (b x + a\right )^{2} \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(-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 \cosh \left (b x + a\right )^{2} \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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