Optimal. Leaf size=33 \[ \frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.0543393, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5450, 3310, 30, 3720, 3475} \[ \frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3310
Rule 30
Rule 3720
Rule 3475
Rubi steps
\begin{align*} \int x \cosh ^2(x) \coth ^2(x) \, dx &=\int x \cosh ^2(x) \, dx+\int x \coth ^2(x) \, dx\\ &=-\frac{1}{4} \cosh ^2(x)-x \coth (x)+\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\int x \, dx}{2}+\int x \, dx+\int \coth (x) \, dx\\ &=\frac{3 x^2}{4}-\frac{\cosh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))+\frac{1}{2} x \cosh (x) \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0288381, size = 33, normalized size = 1. \[ \frac{3 x^2}{4}+\frac{1}{4} x \sinh (2 x)-\frac{1}{8} \cosh (2 x)-x \coth (x)+\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 48, normalized size = 1.5 \begin{align*}{\frac{3\,{x}^{2}}{4}}+ \left ( -{\frac{1}{16}}+{\frac{x}{8}} \right ){{\rm e}^{2\,x}}+ \left ( -{\frac{1}{16}}-{\frac{x}{8}} \right ){{\rm e}^{-2\,x}}-2\,x-2\,{\frac{x}{{{\rm e}^{2\,x}}-1}}+\ln \left ({{\rm e}^{2\,x}}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15228, size = 1022, normalized size = 30.97 \begin{align*} \frac{{\left (2 \, x - 1\right )} \cosh \left (x\right )^{6} + 6 \,{\left (2 \, x - 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} +{\left (2 \, x - 1\right )} \sinh \left (x\right )^{6} +{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{4} +{\left (15 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{2} + 12 \, x^{2} - 34 \, x + 1\right )} \sinh \left (x\right )^{4} + 4 \,{\left (5 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{3} +{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} -{\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )^{2} +{\left (15 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{4} + 6 \,{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{2} - 12 \, x^{2} - 2 \, x - 1\right )} \sinh \left (x\right )^{2} + 16 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (3 \,{\left (2 \, x - 1\right )} \cosh \left (x\right )^{5} + 2 \,{\left (12 \, x^{2} - 34 \, x + 1\right )} \cosh \left (x\right )^{3} -{\left (12 \, x^{2} + 2 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 2 \, x + 1}{16 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19143, size = 136, normalized size = 4.12 \begin{align*} \frac{12 \, x^{2} e^{\left (4 \, x\right )} - 12 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (6 \, x\right )} - 34 \, x e^{\left (4 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + 16 \, e^{\left (4 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 16 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, x - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}{16 \,{\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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