Optimal. Leaf size=101 \[ \frac{4 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac{2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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Rubi [A] time = 0.249677, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5511, 2251} \[ \frac{4 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac{2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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Rule 5511
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \coth (a+b x) \text{csch}(a+b x) \, dx &=\int \left (\frac{4 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+4 \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{4 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}\\ \end{align*}
Mathematica [A] time = 0.656673, size = 92, normalized size = 0.91 \[ \frac{e^c \text{csch}(a) \left (-2 d e^{x (b+d)} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )-(b+d) e^{d x} (\cosh (a)-\sinh (a)) \text{csch}(a+b x)\right )}{b (\coth (a)-1) (b+d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}}\cosh \left ( bx+a \right ) \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 (b x + d x + a + c\right )}}{3 \, b^{2} - 4 \, b d + d^{2} -{\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} - \frac{2 \,{\left ({\left (3 \, b e^{c} - d e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} -{\left (3 \, b e^{c} + d e^{c}\right )} e^{\left (b x + a\right )}\right )} e^{\left (d x\right )}}{3 \, b^{2} - 4 \, b d + d^{2} +{\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (3 \, b^{2} - 4 \, 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 (\cosh \left (b x + a\right ) \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 ) \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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