Optimal. Leaf size=9 \[ x \cosh (x)-\sinh (x) \]
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Rubi [A] time = 0.0109491, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3296, 2637} \[ x \cosh (x)-\sinh (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \sinh (x) \, dx &=x \cosh (x)-\int \cosh (x) \, dx\\ &=x \cosh (x)-\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0023825, size = 9, normalized size = 1. \[ x \cosh (x)-\sinh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 10, normalized size = 1.1 \begin{align*} x\cosh \left ( x \right ) -\sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.951741, size = 46, normalized size = 5.11 \begin{align*} \frac{1}{2} \, x^{2} \sinh \left (x\right ) + \frac{1}{4} \,{\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac{1}{4} \,{\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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Fricas [A] time = 1.56237, size = 28, normalized size = 3.11 \begin{align*} x \cosh \left (x\right ) - \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.172455, size = 7, normalized size = 0.78 \begin{align*} x \cosh{\left (x \right )} - \sinh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08537, size = 23, normalized size = 2.56 \begin{align*} \frac{1}{2} \,{\left (x + 1\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x - 1\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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