Optimal. Leaf size=135 \[ -\frac{6 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{12 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{8 e^{c+d x} \, _2F_1\left (3,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d} \]
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Rubi [A] time = 0.164018, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5485, 2194, 2251} \[ -\frac{6 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{12 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{8 e^{c+d x} \, _2F_1\left (3,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 5485
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \coth ^3(a+b x) \, dx &=\int \left (e^{c+d x}+\frac{8 e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^3}+\frac{12 e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^2}+\frac{6 e^{c+d x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=6 \int \frac{e^{c+d x}}{-1+e^{2 (a+b x)}} \, dx+8 \int \frac{e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^3} \, dx+12 \int \frac{e^{c+d x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+\int e^{c+d x} \, dx\\ &=\frac{e^{c+d x}}{d}-\frac{6 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{12 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};1+\frac{d}{2 b};e^{2 (a+b x)}\right )}{d}-\frac{8 e^{c+d x} \, _2F_1\left (3,\frac{d}{2 b};1+\frac{d}{2 b};e^{2 (a+b x)}\right )}{d}\\ \end{align*}
Mathematica [A] time = 3.72413, size = 176, normalized size = 1.3 \[ \frac{1}{2} e^c \left (-\frac{2 e^{2 a} \left (2 b^2+d^2\right ) \left (\frac{e^{d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{e^{x (2 b+d)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{\left (e^{2 a}-1\right ) b^2}+\frac{d \text{csch}(a) e^{d x} \sinh (b x) \text{csch}(a+b x)}{b^2}-\frac{e^{d x} \text{csch}^2(a+b x)}{b}+\frac{2 \coth (a) e^{d x}}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.176, 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 ) ^{3}\, 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*} -48 \,{\left (2 \, b^{3} e^{c} + b d^{2} e^{c}\right )} \int \frac{e^{\left (d x\right )}}{48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3} +{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{{\left (48 \, b^{3} e^{c} + 44 \, b^{2} d e^{c} + 36 \, b d^{2} e^{c} + d^{3} e^{c} -{\left (48 \, b^{3} e^{c} - 44 \, b^{2} d e^{c} + 12 \, b d^{2} e^{c} - d^{3} e^{c}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (48 \, b^{3} e^{c} + 4 \, b^{2} d e^{c} - 8 \, b d^{2} e^{c} + d^{3} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (48 \, b^{3} e^{c} + 28 \, b^{2} d e^{c} - d^{3} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{48 \, b^{3} d - 44 \, b^{2} d^{2} + 12 \, b d^{3} - d^{4} -{\left (48 \, b^{3} d - 44 \, b^{2} d^{2} + 12 \, b d^{3} - d^{4}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (48 \, b^{3} d - 44 \, b^{2} d^{2} + 12 \, b d^{3} - d^{4}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (48 \, b^{3} d - 44 \, b^{2} d^{2} + 12 \, b d^{3} - d^{4}\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 )^{3} \operatorname{csch}\left (b x + a\right )^{3} 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 )^{3} e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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