Optimal. Leaf size=29 \[ \frac {d \log (\sinh (a+b x))}{b^2}-\frac {(c+d x) \coth (a+b x)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4184, 3475} \[ \frac {d \log (\sinh (a+b x))}{b^2}-\frac {(c+d x) \coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rubi steps
\begin {align*} \int (c+d x) \text {csch}^2(a+b x) \, dx &=-\frac {(c+d x) \coth (a+b x)}{b}+\frac {d \int \coth (a+b x) \, dx}{b}\\ &=-\frac {(c+d x) \coth (a+b x)}{b}+\frac {d \log (\sinh (a+b x))}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 52, normalized size = 1.79 \[ \frac {d \log (\sinh (a+b x))}{b^2}-\frac {c \coth (a+b x)}{b}-\frac {d x \coth (a)}{b}+\frac {d x \text {csch}(a) \sinh (b x) \text {csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 166, normalized size = 5.72 \[ -\frac {2 \, b d x \cosh \left (b x + a\right )^{2} + 4 \, b d x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, b d x \sinh \left (b x + a\right )^{2} + 2 \, b c - {\left (d \cosh \left (b x + a\right )^{2} + 2 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d \sinh \left (b x + a\right )^{2} - d\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 80, normalized size = 2.76 \[ -\frac {2 \, b d x e^{\left (2 \, b x + 2 \, a\right )} - d e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, b c + d \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 56, normalized size = 1.93 \[ -\frac {2 d x}{b}-\frac {2 d a}{b^{2}}-\frac {2 \left (d x +c \right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {d \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 91, normalized size = 3.14 \[ -d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {2 \, c}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 49, normalized size = 1.69 \[ \frac {d\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b^2}-\frac {2\,\left (c+d\,x\right )}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2\,d\,x}{b} \]
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 \[ \int \left (c + d x\right ) \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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