Optimal. Leaf size=160 \[ \frac{6 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{4 e^{-a-x (b-d)+c} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{5 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)} \]
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Rubi [A] time = 0.305758, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5511, 2194, 2227, 2251} \[ \frac{6 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{4 e^{-a-x (b-d)+c} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{5 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)} \]
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 ^2(a+b x) \, dx &=\int \left (\frac{5}{2} e^{-a+c-(b-d) x}+\frac{1}{2} e^{-a+c-(b-d) x+2 (a+b x)}+\frac{4 e^{-a+c-(b-d) x}}{\left (-1+e^{2 (a+b x)}\right )^2}+\frac{6 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{5}{2} \int e^{-a+c-(b-d) x} \, dx+4 \int \frac{e^{-a+c-(b-d) x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+6 \int \frac{e^{-a+c-(b-d) x}}{-1+e^{2 (a+b x)}} \, dx\\ &=-\frac{5 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{6 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{4 e^{-a+c-(b-d) x} \, _2F_1\left (2,-\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{5 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{e^{a+c+(b+d) x}}{2 (b+d)}+\frac{6 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{4 e^{-a+c-(b-d) x} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}\\ \end{align*}
Mathematica [A] time = 1.20762, size = 145, normalized size = 0.91 \[ \frac{e^{c-\frac{a d}{b}} \text{csch}(a+b x) \left (e^{d \left (\frac{a}{b}+x\right )} \left (b^2 \cosh (2 (a+b x))-b d \sinh (2 (a+b x))-3 b^2+2 d^2\right )-4 d (b-d) e^{\frac{(b+d) (a+b x)}{b}} \sinh (a+b x) \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )\right )}{2 b (b-d) (b+d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.163, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \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 )}}{{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (7 \, b x + 7 \, a\right )} - 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (5 \, b x + 5 \, a\right )} + 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )} -{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (b x + a\right )}}\,{d x} - \frac{{\left (15 \, b^{3} e^{c} + 39 \, b^{2} d e^{c} + 25 \, b d^{2} e^{c} + d^{3} e^{c} -{\left (15 \, b^{3} e^{c} - 23 \, b^{2} d e^{c} + 9 \, b d^{2} e^{c} - d^{3} e^{c}\right )} e^{\left (6 \, b x + 6 \, a\right )} +{\left (105 \, b^{3} e^{c} - 11 \, b^{2} d e^{c} - 17 \, b d^{2} e^{c} + 3 \, d^{3} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} -{\left (105 \, b^{3} e^{c} + 59 \, b^{2} d e^{c} - b d^{2} e^{c} - 3 \, d^{3} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{2 \,{\left ({\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \,{\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )} +{\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\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 )^{3} \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 )^{3} \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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