Optimal. Leaf size=103 \[ \frac{2 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{3 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)} \]
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Rubi [A] time = 0.207046, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5511, 2194, 2227, 2251} \[ \frac{2 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{3 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)} \]
Antiderivative was successfully verified.
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Rule 5511
Rule 2194
Rule 2227
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \cosh (a+b x) \coth (a+b x) \, dx &=\int \left (\frac{3}{2} e^{-a+c-(b-d) x}+\frac{1}{2} e^{-a+c-(b-d) x+2 (a+b x)}+\frac{2 e^{-a+c-(b-d) x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=\frac{1}{2} \int e^{-a+c-(b-d) x+2 (a+b x)} \, dx+\frac{3}{2} \int e^{-a+c-(b-d) x} \, dx+2 \int \frac{e^{-a+c-(b-d) x}}{-1+e^{2 (a+b x)}} \, dx\\ &=-\frac{3 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{2 e^{-a+c-(b-d) x} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}+\frac{1}{2} \int e^{a+c+(b+d) x} \, dx\\ &=-\frac{3 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{e^{a+c+(b+d) x}}{2 (b+d)}+\frac{2 e^{-a+c-(b-d) x} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}\\ \end{align*}
Mathematica [A] time = 0.554539, size = 93, normalized size = 0.9 \[ \frac{e^c \left (\frac{e^{d x} (b \cosh (a+b x)-d \sinh (a+b x))}{b-d}-2 (\sinh (a)+\cosh (a)) e^{x (b+d)} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )\right )}{b+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}{\rm csch} \left (bx+a\right )\, 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*} -4 \, b \int \frac{e^{\left (d x + c\right )}}{{\left (3 \, b - d\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \,{\left (3 \, b - d\right )} e^{\left (3 \, b x + 3 \, a\right )} +{\left (3 \, b - d\right )} e^{\left (b x + a\right )}}\,{d x} + \frac{{\left (5 \, b^{2} e^{c} + 6 \, b d e^{c} + d^{2} e^{c} +{\left (3 \, b^{2} e^{c} - 4 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (6 \, b^{2} e^{c} + b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{2 \,{\left ({\left (3 \, b^{3} - b^{2} d - 3 \, b d^{2} + d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )} -{\left (3 \, b^{3} - b^{2} d - 3 \, b d^{2} + d^{3}\right )} e^{\left (b x + a\right )}\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 ) 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 ) e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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