Optimal. Leaf size=30 \[ \frac {x^3}{3}+\frac {x}{2}+2 x \sinh (x)-2 \cosh (x)+\frac {1}{2} \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, 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, 2638, 2635, 8} \[ \frac {x^3}{3}+\frac {x}{2}+2 x \sinh (x)-2 \cosh (x)+\frac {1}{2} \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 3296
Rule 6742
Rubi steps
\begin {align*} \int (x+\cosh (x))^2 \, dx &=\int \left (x^2+2 x \cosh (x)+\cosh ^2(x)\right ) \, dx\\ &=\frac {x^3}{3}+2 \int x \cosh (x) \, dx+\int \cosh ^2(x) \, dx\\ &=\frac {x^3}{3}+2 x \sinh (x)+\frac {1}{2} \cosh (x) \sinh (x)+\frac {\int 1 \, dx}{2}-2 \int \sinh (x) \, dx\\ &=\frac {x}{2}+\frac {x^3}{3}-2 \cosh (x)+2 x \sinh (x)+\frac {1}{2} \cosh (x) \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 26, normalized size = 0.87 \[ \frac {1}{6} \left (x \left (2 x^2+12 \sinh (x)+3\right )+3 (\sinh (x)-4) \cosh (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 23, normalized size = 0.77 \[ \frac {1}{3} \, x^{3} + \frac {1}{2} \, {\left (4 \, x + \cosh \relax (x)\right )} \sinh \relax (x) + \frac {1}{2} \, x - 2 \, \cosh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 36, normalized size = 1.20 \[ \frac {1}{3} \, x^{3} - {\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} + \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.83 \[ \frac {x}{2}+\frac {x^{3}}{3}-2 \cosh \relax (x )+2 x \sinh \relax (x )+\frac {\cosh \relax (x ) \sinh \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 1.20 \[ \frac {1}{3} \, x^{3} - {\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} + \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.80 \[ \frac {x}{2}-2\,\mathrm {cosh}\relax (x)+\frac {\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2}+2\,x\,\mathrm {sinh}\relax (x)+\frac {x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 41, normalized size = 1.37 \[ \frac {x^{3}}{3} - \frac {x \sinh ^{2}{\relax (x )}}{2} + 2 x \sinh {\relax (x )} + \frac {x \cosh ^{2}{\relax (x )}}{2} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2} - 2 \cosh {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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