Optimal. Leaf size=151 \[ -\frac {2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac {8 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac {8 e^{a+x (b+d)+c} \, _2F_1\left (3,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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Rubi [A] time = 0.34, 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 = {5511, 2251} \[ -\frac {2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}+\frac {8 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac {8 e^{a+x (b+d)+c} \, _2F_1\left (3,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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Rule 2251
Rule 5511
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 = 1.33, size = 111, normalized size = 0.74 \[ -\frac {e^{c-\frac {a d}{b}} \left (2 \left (b^2+d^2\right ) e^{\frac {(b+d) (a+b x)}{b}} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )+(b+d) e^{d \left (\frac {a}{b}+x\right )} \text {csch}(a+b x) (b \coth (a+b x)+d)\right )}{2 b^2 (b+d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.75, 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 \[ -48 \, {\left (b^{3} e^{c} + b d^{2} e^{c}\right )} \int \frac {e^{\left (b x + d x + a\right )}}{15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3} + {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \, {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \, {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac {2 \, {\left ({\left (15 \, b^{2} e^{c} - 8 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \, {\left (10 \, b^{2} e^{c} + 3 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} + {\left (9 \, b^{2} e^{c} + 14 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (b x + a\right )}\right )} e^{\left (d x\right )}}{15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3} - {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \, {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \, {\left (15 \, b^{3} - 23 \, b^{2} d + 9 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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