Optimal. Leaf size=16 \[ x^2 \sinh (x)+2 \sinh (x)-2 x \cosh (x) \]
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Rubi [A] time = 0.0271148, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3296, 2637} \[ x^2 \sinh (x)+2 \sinh (x)-2 x \cosh (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \cosh (x) \, dx &=x^2 \sinh (x)-2 \int x \sinh (x) \, dx\\ &=-2 x \cosh (x)+x^2 \sinh (x)+2 \int \cosh (x) \, dx\\ &=-2 x \cosh (x)+2 \sinh (x)+x^2 \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0129836, size = 14, normalized size = 0.88 \[ \left (x^2+2\right ) \sinh (x)-2 x \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 1.1 \begin{align*} -2\,x\cosh \left ( x \right ) +2\,\sinh \left ( x \right ) +{x}^{2}\sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.95329, size = 59, normalized size = 3.69 \begin{align*} \frac{1}{3} \, x^{3} \cosh \left (x\right ) - \frac{1}{6} \,{\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} - \frac{1}{6} \,{\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8497, size = 46, normalized size = 2.88 \begin{align*} -2 \, x \cosh \left (x\right ) +{\left (x^{2} + 2\right )} \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.326455, size = 17, normalized size = 1.06 \begin{align*} x^{2} \sinh{\left (x \right )} - 2 x \cosh{\left (x \right )} + 2 \sinh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09227, size = 36, normalized size = 2.25 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x^{2} - 2 \, x + 2\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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