Optimal. Leaf size=54 \[ \frac{4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d} \]
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Rubi [A] time = 0.0294736, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5493} \[ \frac{4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d} \]
Antiderivative was successfully verified.
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Rule 5493
Rubi steps
\begin{align*} \int e^{c+d x} \text{csch}^2(a+b x) \, dx &=\frac{4 e^{c+d x+2 (a+b x)} \, _2F_1\left (2,1+\frac{d}{2 b};2+\frac{d}{2 b};e^{2 (a+b x)}\right )}{2 b+d}\\ \end{align*}
Mathematica [B] time = 3.22928, size = 131, normalized size = 2.43 \[ \frac{e^c \left (\text{csch}(a) e^{d x} \sinh (b x) \text{csch}(a+b x)-\frac{2 e^{2 a} d \left (\frac{e^{d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{e^{x (2 b+d)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{e^{2 a}-1}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \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{4 \,{\left ({\left (4 \, b e^{c} - d e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b e^{c}\right )} e^{\left (d x\right )}}{8 \, b^{2} - 6 \, b d + d^{2} +{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} - 6 \, 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 )^{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] time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int e^{d x} \operatorname{csch}^{2}{\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 )^{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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