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 = 0.04, 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, 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 8
Rule 2635
Rule 2637
Rule 3296
Rule 6742
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*}
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Mathematica [A] time = 0.06, size = 30, normalized size = 1.00 \[ \frac {1}{6} x \left (2 x^2-3\right )-2 \sinh (x)+\frac {1}{4} \sinh (2 x)+2 x \cosh (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 22, normalized size = 0.73 \[ \frac {1}{3} \, x^{3} + 2 \, x \cosh \relax (x) + \frac {1}{2} \, {\left (\cosh \relax (x) - 4\right )} \sinh \relax (x) - \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 1.17 \[ \frac {1}{3} \, x^{3} + {\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} - \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 25, normalized size = 0.83 \[ -\frac {x}{2}+\frac {x^{3}}{3}+2 x \cosh \relax (x )-2 \sinh \relax (x )+\frac {\cosh \relax (x ) \sinh \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 35, normalized size = 1.17 \[ \frac {1}{3} \, x^{3} + {\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} - \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 24, normalized size = 0.80 \[ \frac {\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2}-2\,\mathrm {sinh}\relax (x)-\frac {x}{2}+2\,x\,\mathrm {cosh}\relax (x)+\frac {x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 1.37 \[ \frac {x^{3}}{3} + \frac {x \sinh ^{2}{\relax (x )}}{2} - \frac {x \cosh ^{2}{\relax (x )}}{2} + 2 x \cosh {\relax (x )} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2} - 2 \sinh {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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