Optimal. Leaf size=30 \[ \frac{x^3}{3}-\frac{x}{2}-2 \sinh (x)+2 x \cosh (x)+\frac{1}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.0376925, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6742, 3296, 2637, 2635, 8} \[ \frac{x^3}{3}-\frac{x}{2}-2 \sinh (x)+2 x \cosh (x)+\frac{1}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3296
Rule 2637
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (x+\sinh (x))^2 \, dx &=\int \left (x^2+2 x \sinh (x)+\sinh ^2(x)\right ) \, dx\\ &=\frac{x^3}{3}+2 \int x \sinh (x) \, dx+\int \sinh ^2(x) \, dx\\ &=\frac{x^3}{3}+2 x \cosh (x)+\frac{1}{2} \cosh (x) \sinh (x)-\frac{\int 1 \, dx}{2}-2 \int \cosh (x) \, dx\\ &=-\frac{x}{2}+\frac{x^3}{3}+2 x \cosh (x)-2 \sinh (x)+\frac{1}{2} \cosh (x) \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0555372, size = 30, normalized size = 1. \[ \frac{1}{6} x \left (2 x^2-3\right )-2 \sinh (x)+\frac{1}{4} \sinh (2 x)+2 x \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 25, normalized size = 0.8 \begin{align*} -{\frac{x}{2}}+{\frac{{x}^{3}}{3}}+2\,x\cosh \left ( x \right ) -2\,\sinh \left ( x \right ) +{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07062, size = 47, normalized size = 1.57 \begin{align*} \frac{1}{3} \, x^{3} +{\left (x + 1\right )} e^{\left (-x\right )} +{\left (x - 1\right )} e^{x} - \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8074, size = 80, normalized size = 2.67 \begin{align*} \frac{1}{3} \, x^{3} + 2 \, x \cosh \left (x\right ) + \frac{1}{2} \,{\left (\cosh \left (x\right ) - 4\right )} \sinh \left (x\right ) - \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.202045, size = 41, normalized size = 1.37 \begin{align*} \frac{x^{3}}{3} + \frac{x \sinh ^{2}{\left (x \right )}}{2} - \frac{x \cosh ^{2}{\left (x \right )}}{2} + 2 x \cosh{\left (x \right )} + \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{2} - 2 \sinh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29417, size = 47, normalized size = 1.57 \begin{align*} \frac{1}{3} \, x^{3} +{\left (x + 1\right )} e^{\left (-x\right )} +{\left (x - 1\right )} e^{x} - \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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