Optimal. Leaf size=47 \[ -\frac{\cosh (a+b x)}{b^2}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}+\frac{x \sinh (a+b x)}{b}-\frac{x \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.0541248, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5450, 3296, 2638, 5419, 3770} \[ -\frac{\cosh (a+b x)}{b^2}-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}+\frac{x \sinh (a+b x)}{b}-\frac{x \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3296
Rule 2638
Rule 5419
Rule 3770
Rubi steps
\begin{align*} \int x \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x \cosh (a+b x) \, dx+\int x \coth (a+b x) \text{csch}(a+b x) \, dx\\ &=-\frac{x \text{csch}(a+b x)}{b}+\frac{x \sinh (a+b x)}{b}+\frac{\int \text{csch}(a+b x) \, dx}{b}-\frac{\int \sinh (a+b x) \, dx}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac{\cosh (a+b x)}{b^2}-\frac{x \text{csch}(a+b x)}{b}+\frac{x \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.218691, size = 66, normalized size = 1.4 \[ \frac{2 b x \sinh (a+b x)-2 \cosh (a+b x)+b x \tanh \left (\frac{1}{2} (a+b x)\right )-b x \coth \left (\frac{1}{2} (a+b x)\right )+2 \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 89, normalized size = 1.9 \begin{align*}{\frac{ \left ( bx-1 \right ){{\rm e}^{bx+a}}}{2\,{b}^{2}}}-{\frac{ \left ( bx+1 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{2}}}-2\,{\frac{{{\rm e}^{bx+a}}x}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.36774, size = 147, normalized size = 3.13 \begin{align*} -\frac{6 \, b x e^{\left (b x + 2 \, a\right )} -{\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} -{\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \,{\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} - b^{2} e^{a}\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 = 2.29503, size = 1008, normalized size = 21.45 \begin{align*} \frac{{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \,{\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} - 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \,{\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x - 2 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} +{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} +{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \,{\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} - 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \,{\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right ) +{\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23585, size = 194, normalized size = 4.13 \begin{align*} \frac{b x e^{\left (4 \, b x + 4 \, a\right )} - 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \,{\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} - b^{2} e^{\left (b x + a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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