Optimal. Leaf size=125 \[ -\frac {7 e^{-2 a+c-(2 b-d) x}}{4 (2 b-d)}+\frac {e^{c+d x}}{d}+\frac {e^{2 a+c+(2 b+d) x}}{4 (2 b+d)}+\frac {2 e^{-2 a+c-(2 b-d) x} \, _2F_1\left (1,\frac {1}{2} \left (-2+\frac {d}{b}\right );\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b-d} \]
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Rubi [A]
time = 0.17, 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 = {5622, 2225,
2259, 2283} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2259
Rule 2283
Rule 5622
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 = 0.76, size = 172, normalized size = 1.38 \begin {gather*} -\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};1+\frac {d}{2 b};e^{2 (a+b x)}\right )+2 (2 b-d) d e^{\left (2+\frac {d}{b}\right ) (a+b x)} \, _2F_1\left (1,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )+d e^{d \left (\frac {a}{b}+x\right )} (-2 b \cosh (2 (a+b x))+d \sinh (2 (a+b x)))\right )}{8 b^2 d-2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.57, 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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^3\,{\mathrm {e}}^{c+d\,x}}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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