Optimal. Leaf size=161 \[ -\frac{6 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac{12 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}-\frac{8 e^{c (a+b x)} \, _2F_1\left (3,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac{e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.175779, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5485, 2194, 2251} \[ -\frac{6 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac{12 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}-\frac{8 e^{c (a+b x)} \, _2F_1\left (3,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac{e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 5485
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int e^{c (a+b x)} \coth ^3(d+e x) \, dx &=\int \left (e^{c (a+b x)}+\frac{8 e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^3}+\frac{12 e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2}+\frac{6 e^{c (a+b x)}}{-1+e^{2 (d+e x)}}\right ) \, dx\\ &=6 \int \frac{e^{c (a+b x)}}{-1+e^{2 (d+e x)}} \, dx+8 \int \frac{e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^3} \, dx+12 \int \frac{e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac{e^{c (a+b x)}}{b c}-\frac{6 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};1+\frac{b c}{2 e};e^{2 (d+e x)}\right )}{b c}+\frac{12 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};1+\frac{b c}{2 e};e^{2 (d+e x)}\right )}{b c}-\frac{8 e^{c (a+b x)} \, _2F_1\left (3,\frac{b c}{2 e};1+\frac{b c}{2 e};e^{2 (d+e x)}\right )}{b c}\\ \end{align*}
Mathematica [A] time = 3.31338, size = 185, normalized size = 1.15 \[ \frac{1}{2} e^{c (a+b x)} \left (\frac{2 e^{2 d} \left (b^2 c^2+2 e^2\right ) \left (b c e^{2 e x} \, _2F_1\left (1,\frac{b c}{2 e}+1;\frac{b c}{2 e}+2;e^{2 (d+e x)}\right )-(b c+2 e) \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;e^{2 (d+e x)}\right )\right )}{b c \left (e^{2 d}-1\right ) e^2 (b c+2 e)}+\frac{b c \text{csch}(d) \sinh (e x) \text{csch}(d+e x)}{e^2}+\frac{2 \coth (d)}{b c}-\frac{\text{csch}^2(d+e x)}{e}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c \left ( bx+a \right ) }} \left ({\rm coth} \left (ex+d\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*} \text{result too large to display} \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 (\coth \left (e x + d\right )^{3} e^{\left (b c x + a 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 \coth \left (e x + d\right )^{3} e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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