Optimal. Leaf size=50 \[ -\frac{2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );e^{2 (c+d x)}\right )}{b+d} \]
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Rubi [A] time = 0.0191584, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5493} \[ -\frac{2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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Rule 5493
Rubi steps
\begin{align*} \int e^{a+b x} \text{csch}(c+d x) \, dx &=-\frac{2 e^{a+c+b x+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (3+\frac{b}{d}\right );e^{2 (c+d x)}\right )}{b+d}\\ \end{align*}
Mathematica [A] time = 0.132807, size = 59, normalized size = 1.18 \[ -\frac{2 (\sinh (c)+\cosh (c)) e^{a+x (b+d)} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{b+3 d}{2 d};e^{2 d x} (\cosh (c)+\sinh (c))^2\right )}{b+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}}{\rm csch} \left (dx+c\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (d x + c\right ) e^{\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{csch}\left (d x + c\right ) e^{\left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int e^{b x} \operatorname{csch}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (d x + c\right ) e^{\left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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