Optimal. Leaf size=151 \[ -\frac {2 e^{a+c+(b+d) x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac {8 e^{a+c+(b+d) x} \, _2F_1\left (2,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac {8 e^{a+c+(b+d) x} \, _2F_1\left (3,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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Rubi [A]
time = 0.25, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5622, 2283}
\begin {gather*} -\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}+\frac {8 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 {8 e^{a+x (b+d)+c} \, _2F_1\left (3,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2283
Rule 5622
Rubi steps
\begin {align*} \int e^{c+d x} \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\int \left (\frac {8 e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3}+\frac {8 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+8 \int \frac {e^{a+c+(b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3} \, dx+8 \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 {8 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}-\frac {8 e^{a+c+(b+d) x} \, _2F_1\left (3,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}\\ \end {align*}
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Mathematica [A]
time = 0.91, size = 111, normalized size = 0.74 \begin {gather*} -\frac {e^{c-\frac {a d}{b}} \left ((b+d) e^{d \left (\frac {a}{b}+x\right )} (d+b \coth (a+b x)) \text {csch}(a+b x)+2 \left (b^2+d^2\right ) e^{\frac {(b+d) (a+b x)}{b}} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )\right )}{2 b^2 (b+d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.50, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d x +c} \left (\cosh ^{2}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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