Optimal. Leaf size=101 \[ \frac {4 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac {2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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Rubi [A] time = 0.25, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5511, 2251} \[ \frac {4 e^{a+x (b+d)+c} \, _2F_1\left (2,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d}-\frac {2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 (a+b x)}\right )}{b+d} \]
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}(a+b x) \, dx &=\int \left (\frac {4 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+4 \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 {4 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}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 92, normalized size = 0.91 \[ \frac {e^c \text {csch}(a) \left (-2 d e^{x (b+d)} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )-(b+d) e^{d x} (\cosh (a)-\sinh (a)) \text {csch}(a+b x)\right )}{b (\coth (a)-1) (b+d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \cosh \left (b x +a \right ) \mathrm {csch}\left (b x +a \right )^{2}\, dx \]
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 \[ 16 \, b d \int -\frac {e^{\left (b x + d x + a + c\right )}}{3 \, b^{2} - 4 \, b d + d^{2} - {\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \, {\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \, {\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} - \frac {2 \, {\left ({\left (3 \, b e^{c} - d e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} - {\left (3 \, b e^{c} + d e^{c}\right )} e^{\left (b x + a\right )}\right )} e^{\left (d x\right )}}{3 \, b^{2} - 4 \, b d + d^{2} + {\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, {\left (3 \, b^{2} - 4 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]
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 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
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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