Optimal. Leaf size=31 \[ \frac {\log (\sinh (a+b x))}{b^2}-\frac {x \coth (a+b x)}{b}+\frac {x^2}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3720, 3475, 30} \[ \frac {\log (\sinh (a+b x))}{b^2}-\frac {x \coth (a+b x)}{b}+\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3475
Rule 3720
Rubi steps
\begin {align*} \int x \coth ^2(a+b x) \, dx &=-\frac {x \coth (a+b x)}{b}+\frac {\int \coth (a+b x) \, dx}{b}+\int x \, dx\\ &=\frac {x^2}{2}-\frac {x \coth (a+b x)}{b}+\frac {\log (\sinh (a+b x))}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 46, normalized size = 1.48 \[ \frac {-2 b x \coth (a)+2 \log (\sinh (a+b x))+2 b x \text {csch}(a) \sinh (b x) \text {csch}(a+b x)+b^2 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 189, normalized size = 6.10 \[ -\frac {b^{2} x^{2} - {\left (b^{2} x^{2} - 4 \, b x\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 4 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} - 4 \, b x\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 98, normalized size = 3.16 \[ \frac {b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2} x^{2} - 4 \, b x e^{\left (2 \, b x + 2 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) - 2 \, \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{2 \, {\left (b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 54, normalized size = 1.74 \[ \frac {x^{2}}{2}-\frac {2 x}{b}-\frac {2 a}{b^{2}}-\frac {2 x}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 115, normalized size = 3.71 \[ -\frac {x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {b x^{2} - {\left (b x^{2} e^{\left (2 \, a\right )} - 2 \, x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{2 \, {\left (b e^{\left (2 \, b x + 2 \, a\right )} - b\right )}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 45, normalized size = 1.45 \[ \frac {\frac {x^2\,\mathrm {sinh}\left (a+b\,x\right )}{2}-\frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{b}}{\mathrm {sinh}\left (a+b\,x\right )}+\frac {\ln \left (\mathrm {sinh}\left (a+b\,x\right )\right )}{b^2} \]
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 \[ \int x \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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