Optimal. Leaf size=54 \[ \frac {4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d} \]
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Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5493} \[ \frac {4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d} \]
Antiderivative was successfully verified.
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Rule 5493
Rubi steps
\begin {align*} \int e^{c+d x} \text {csch}^2(a+b x) \, dx &=\frac {4 e^{c+d x+2 (a+b x)} \, _2F_1\left (2,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b+d}\\ \end {align*}
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Mathematica [B] time = 3.37, size = 131, normalized size = 2.43 \[ \frac {e^c \left (\text {csch}(a) e^{d x} \sinh (b x) \text {csch}(a+b x)-\frac {2 e^{2 a} d \left (\frac {e^{d x} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac {e^{x (2 b+d)} \, _2F_1\left (1,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{e^{2 a}-1}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \operatorname {csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \mathrm {csch}\left (b x +a \right )^{2}\, dx \]
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 \[ 16 \, b d \int -\frac {e^{\left (d x + c\right )}}{8 \, b^{2} - 6 \, b d + d^{2} - {\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \, {\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \, {\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} - \frac {4 \, {\left ({\left (4 \, b e^{c} - d e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b e^{c}\right )} e^{\left (d x\right )}}{8 \, b^{2} - 6 \, b d + d^{2} + {\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, {\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int e^{d x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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