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 = 0.0267439, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3720
Rule 3475
Rule 30
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*}
Mathematica [A] time = 0.153167, 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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Maple [A] time = 0.029, size = 54, normalized size = 1.7 \begin{align*}{\frac{{x}^{2}}{2}}-2\,{\frac{x}{b}}-2\,{\frac{a}{{b}^{2}}}-2\,{\frac{x}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.184, size = 155, normalized size = 5. \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14701, size = 477, normalized size = 15.39 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.73998, size = 80, normalized size = 2.58 \begin{align*} \begin{cases} \tilde{\infty } x^{2} + \frac{\tilde{\infty } x}{b} & \text{for}\: a = \log{\left (- e^{- b x} \right )} \\\tilde{\infty } x^{2} & \text{for}\: a = \log{\left (e^{- b x} \right )} \\\frac{x^{2} \coth ^{2}{\left (a \right )}}{2} & \text{for}\: b = 0 \\\frac{x^{2}}{2} + \frac{x}{b} - \frac{x}{b \tanh{\left (a + b x \right )}} - \frac{\log{\left (\tanh{\left (a + b x \right )} + 1 \right )}}{b^{2}} + \frac{\log{\left (\tanh{\left (a + b x \right )} \right )}}{b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17756, size = 132, normalized size = 4.26 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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