Optimal. Leaf size=94 \[ -\frac{4 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{4 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d} \]
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Rubi [A] time = 0.114463, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5485, 2194, 2251} \[ -\frac{4 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{4 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 5485
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \coth ^2(a+b x) \, dx &=\int \left (e^{c+d x}+\frac{4 e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^2}+\frac{4 e^{c+d x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=4 \int \frac{e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+4 \int \frac{e^{c+d x}}{-1+e^{2 (a+b x)}} \, dx+\int e^{c+d x} \, dx\\ &=\frac{e^{c+d x}}{d}-\frac{4 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};1+\frac{d}{2 b};e^{2 (a+b x)}\right )}{d}+\frac{4 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};1+\frac{d}{2 b};e^{2 (a+b x)}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.87354, size = 145, normalized size = 1.54 \[ -\frac{2 d e^{2 a+c} \left (\frac{e^{d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{e^{x (2 b+d)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{\left (e^{2 a}-1\right ) b}+\frac{\text{csch}(a) \sinh (b x) e^{c+d x} \text{csch}(a+b x)}{b}+\frac{e^{c+d x}}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, 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 ) ^{2}\, 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*} 16 \, b d \int -\frac{e^{\left (d x + c\right )}}{8 \, b^{2} - 6 \, b d + d^{2} -{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{{\left (8 \, b^{2} e^{c} + 10 \, b d e^{c} + d^{2} e^{c} +{\left (8 \, b^{2} e^{c} - 6 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} e^{c} + 2 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{8 \, b^{2} d - 6 \, b d^{2} + d^{3} +{\left (8 \, b^{2} d - 6 \, b d^{2} + d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} d - 6 \, 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 )^{2} 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 )^{2} e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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