Optimal. Leaf size=113 \[ \frac {4 e^{2 a+x (2 b+d)+c} \, _2F_1\left (2,\frac {1}{2} \left (\frac {d}{b}+2\right );\frac {1}{2} \left (\frac {d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac {8 e^{2 a+x (2 b+d)+c} \, _2F_1\left (3,\frac {1}{2} \left (\frac {d}{b}+2\right );\frac {1}{2} \left (\frac {d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d} \]
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Rubi [A] time = 0.27, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5511, 2251} \[ \frac {4 e^{2 a+x (2 b+d)+c} \, _2F_1\left (2,\frac {1}{2} \left (\frac {d}{b}+2\right );\frac {1}{2} \left (\frac {d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac {8 e^{2 a+x (2 b+d)+c} \, _2F_1\left (3,\frac {1}{2} \left (\frac {d}{b}+2\right );\frac {1}{2} \left (\frac {d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d} \]
Antiderivative was successfully verified.
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Rule 2251
Rule 5511
Rubi steps
\begin {align*} \int e^{c+d x} \coth (a+b x) \text {csch}^2(a+b x) \, dx &=\int \left (\frac {8 e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3}+\frac {4 e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2}\right ) \, dx\\ &=4 \int \frac {e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+8 \int \frac {e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3} \, dx\\ &=\frac {4 e^{2 a+c+(2 b+d) x} \, _2F_1\left (2,\frac {1}{2} \left (2+\frac {d}{b}\right );\frac {1}{2} \left (4+\frac {d}{b}\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac {8 e^{2 a+c+(2 b+d) x} \, _2F_1\left (3,\frac {1}{2} \left (2+\frac {d}{b}\right );\frac {1}{2} \left (4+\frac {d}{b}\right );e^{2 (a+b x)}\right )}{2 b+d}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 159, normalized size = 1.41 \[ -\frac {e^{c-\frac {a d}{b}} \left (d^2 e^{\left (\frac {d}{b}+2\right ) (a+b x)} \, _2F_1\left (1,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )+d (2 b+d) e^{d \left (\frac {a}{b}+x\right )} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )+(2 b+d) e^{d \left (\frac {a}{b}+x\right )} \left (d \coth (a+b x)+b \text {csch}^2(a+b x)\right )\right )}{2 b^2 (2 b+d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \cosh \left (b x +a \right ) \mathrm {csch}\left (b x +a \right )^{3}\, dx \]
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 \[ -48 \, b d^{2} \int \frac {e^{\left (d x + c\right )}}{48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3} + {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \, {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \, {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac {4 \, {\left (12 \, b d e^{c} + {\left (24 \, b^{2} e^{c} - 10 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - {\left (24 \, b^{2} e^{c} + 2 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3} - {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \, {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \, {\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]
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 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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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