Optimal. Leaf size=125 \[ \frac {2 e^{-2 a-x (2 b-d)+c} \, _2F_1\left (1,\frac {1}{2} \left (\frac {d}{b}-2\right );\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}-\frac {7 e^{-2 a-x (2 b-d)+c}}{4 (2 b-d)}+\frac {e^{2 a+x (2 b+d)+c}}{4 (2 b+d)}+\frac {e^{c+d x}}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5511, 2194, 2227, 2251} \[ \frac {2 e^{-2 a-x (2 b-d)+c} \, _2F_1\left (1,\frac {1}{2} \left (\frac {d}{b}-2\right );\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}-\frac {7 e^{-2 a-x (2 b-d)+c}}{4 (2 b-d)}+\frac {e^{2 a+x (2 b+d)+c}}{4 (2 b+d)}+\frac {e^{c+d x}}{d} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2227
Rule 2251
Rule 5511
Rubi steps
\begin {align*} \int e^{c+d x} \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int \left (\frac {7}{4} e^{-2 a+c-(2 b-d) x}+e^{-2 a+c-(2 b-d) x+2 (a+b x)}+\frac {1}{4} e^{-2 a+c-(2 b-d) x+4 (a+b x)}+\frac {2 e^{-2 a+c-(2 b-d) x}}{-1+e^{2 (a+b x)}}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 a+c-(2 b-d) x+4 (a+b x)} \, dx+\frac {7}{4} \int e^{-2 a+c-(2 b-d) x} \, dx+2 \int \frac {e^{-2 a+c-(2 b-d) x}}{-1+e^{2 (a+b x)}} \, dx+\int e^{-2 a+c-(2 b-d) x+2 (a+b x)} \, dx\\ &=-\frac {7 e^{-2 a+c-(2 b-d) x}}{4 (2 b-d)}+\frac {2 e^{-2 a+c-(2 b-d) x} \, _2F_1\left (1,\frac {1}{2} \left (-2+\frac {d}{b}\right );\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}+\frac {1}{4} \int e^{2 a+c+(2 b+d) x} \, dx+\int e^{c+d x} \, dx\\ &=-\frac {7 e^{-2 a+c-(2 b-d) x}}{4 (2 b-d)}+\frac {e^{c+d x}}{d}+\frac {e^{2 a+c+(2 b+d) x}}{4 (2 b+d)}+\frac {2 e^{-2 a+c-(2 b-d) x} \, _2F_1\left (1,\frac {1}{2} \left (-2+\frac {d}{b}\right );\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b-d}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 172, normalized size = 1.38 \[ -\frac {e^{c-\frac {a d}{b}} \left (2 \left (4 b^2-d^2\right ) e^{d \left (\frac {a}{b}+x\right )} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )+2 d (2 b-d) e^{\left (\frac {d}{b}+2\right ) (a+b x)} \, _2F_1\left (1,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )+d e^{d \left (\frac {a}{b}+x\right )} (d \sinh (2 (a+b x))-2 b \cosh (2 (a+b x)))\right )}{8 b^2 d-2 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )\, dx \]
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 \[ -4 \, b \int \frac {e^{\left (d x + c\right )}}{{\left (4 \, b - d\right )} e^{\left (6 \, b x + 6 \, a\right )} - 2 \, {\left (4 \, b - d\right )} e^{\left (4 \, b x + 4 \, a\right )} + {\left (4 \, b - d\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac {{\left (24 \, b^{2} d e^{c} + 14 \, b d^{2} e^{c} + d^{3} e^{c} + {\left (8 \, b^{2} d e^{c} - 6 \, b d^{2} e^{c} + d^{3} e^{c}\right )} e^{\left (6 \, b x + 6 \, a\right )} + {\left (64 \, b^{3} e^{c} - 24 \, b^{2} d e^{c} - 10 \, b d^{2} e^{c} + 3 \, d^{3} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - {\left (64 \, b^{3} e^{c} + 40 \, b^{2} d e^{c} - 2 \, b d^{2} e^{c} - 3 \, d^{3} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{4 \, {\left ({\left (16 \, b^{3} d - 4 \, b^{2} d^{2} - 4 \, b d^{3} + d^{4}\right )} e^{\left (4 \, b x + 4 \, a\right )} - {\left (16 \, b^{3} d - 4 \, b^{2} d^{2} - 4 \, b d^{3} + d^{4}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )}} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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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