Optimal. Leaf size=100 \[ \frac {(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (\frac {d}{b}+3\right );e^{2 (a+b x)}\right )}{b^2}-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5491, 5493} \[ \frac {(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (\frac {d}{b}+3\right );e^{2 (a+b x)}\right )}{b^2}-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5491
Rule 5493
Rubi steps
\begin {align*} \int e^{c+d x} \text {csch}^3(a+b x) \, dx &=-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \left (1-\frac {d^2}{b^2}\right ) \int e^{c+d x} \text {csch}(a+b x) \, dx\\ &=-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (3+\frac {d}{b}\right );e^{2 (a+b x)}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 2.49, size = 94, normalized size = 0.94 \[ \frac {e^c \left (\frac {2 \text {csch}(a) (b-d) e^{x (b+d)} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )}{\coth (a)-1}-e^{d x} \text {csch}(a+b x) (b \coth (a+b x)+d)\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}, x\right ) \]
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 \[ \int \operatorname {csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \mathrm {csch}\left (b x +a \right )^{3}\, dx \]
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 \[ 48 \, {\left (b^{2} e^{c} + b d e^{c}\right )} \int \frac {e^{\left (b x + d x + a\right )}}{15 \, b^{2} - 8 \, b d + d^{2} + {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \, {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \, {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \, {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac {8 \, {\left ({\left (5 \, b e^{c} - d e^{c}\right )} e^{\left (3 \, b x + 3 \, a\right )} - 6 \, b e^{\left (b x + a + c\right )}\right )} e^{\left (d x\right )}}{15 \, b^{2} - 8 \, b d + d^{2} - {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \, {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \, {\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{c+d\,x}}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]
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 \[ e^{c} \int e^{d x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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