Optimal. Leaf size=50 \[ -\frac {2 e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (3+\frac {b}{d}\right );e^{2 (c+d x)}\right )}{b+d} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {5601}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 5601
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*}
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Mathematica [A]
time = 0.09, size = 59, normalized size = 1.18 \begin {gather*} -\frac {2 e^{a+(b+d) x} \, _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 ) (\cosh (c)+\sinh (c))}{b+d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.41, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{b x +a} \mathrm {csch}\left (d x +c \right )\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a} \int e^{b x} \operatorname {csch}{\left (c + d x \right )}\, dx \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 = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{a+b\,x}}{\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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