Optimal. Leaf size=56 \[ \frac {x^4}{4}+\frac {3 x^2}{4}+3 x^2 \sinh (x)+\frac {\sinh ^3(x)}{3}+7 \sinh (x)-\frac {3 \cosh ^2(x)}{4}-6 x \cosh (x)+\frac {3}{2} x \sinh (x) \cosh (x) \]
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Rubi [A] time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6742, 3296, 2637, 3310, 30, 2633} \[ \frac {x^4}{4}+\frac {3 x^2}{4}+3 x^2 \sinh (x)+\frac {\sinh ^3(x)}{3}+7 \sinh (x)-\frac {3 \cosh ^2(x)}{4}-6 x \cosh (x)+\frac {3}{2} x \sinh (x) \cosh (x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2633
Rule 2637
Rule 3296
Rule 3310
Rule 6742
Rubi steps
\begin {align*} \int (x+\cosh (x))^3 \, dx &=\int \left (x^3+3 x^2 \cosh (x)+3 x \cosh ^2(x)+\cosh ^3(x)\right ) \, dx\\ &=\frac {x^4}{4}+3 \int x^2 \cosh (x) \, dx+3 \int x \cosh ^2(x) \, dx+\int \cosh ^3(x) \, dx\\ &=\frac {x^4}{4}-\frac {3 \cosh ^2(x)}{4}+3 x^2 \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)+i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )+\frac {3 \int x \, dx}{2}-6 \int x \sinh (x) \, dx\\ &=\frac {3 x^2}{4}+\frac {x^4}{4}-6 x \cosh (x)-\frac {3 \cosh ^2(x)}{4}+\sinh (x)+3 x^2 \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)+\frac {\sinh ^3(x)}{3}+6 \int \cosh (x) \, dx\\ &=\frac {3 x^2}{4}+\frac {x^4}{4}-6 x \cosh (x)-\frac {3 \cosh ^2(x)}{4}+7 \sinh (x)+3 x^2 \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)+\frac {\sinh ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 51, normalized size = 0.91 \[ \frac {1}{12} \left (3 x^4+9 x^2+9 \left (4 x^2+9\right ) \sinh (x)+9 x \sinh (2 x)+\sinh (3 x)\right )-6 x \cosh (x)-\frac {3}{8} \cosh (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 54, normalized size = 0.96 \[ \frac {1}{4} \, x^{4} + \frac {1}{12} \, \sinh \relax (x)^{3} + \frac {3}{4} \, x^{2} - 6 \, x \cosh \relax (x) - \frac {3}{8} \, \cosh \relax (x)^{2} + \frac {1}{4} \, {\left (12 \, x^{2} + 6 \, x \cosh \relax (x) + \cosh \relax (x)^{2} + 27\right )} \sinh \relax (x) - \frac {3}{8} \, \sinh \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 75, normalized size = 1.34 \[ \frac {1}{4} \, x^{4} + \frac {3}{4} \, x^{2} + \frac {3}{16} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {3}{8} \, {\left (4 \, x^{2} + 8 \, x + 9\right )} e^{\left (-x\right )} - \frac {3}{16} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{8} \, {\left (4 \, x^{2} - 8 \, x + 9\right )} e^{x} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 52, normalized size = 0.93 \[ \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\relax (x )\right )}{3}\right ) \sinh \relax (x )+\frac {3 x \cosh \relax (x ) \sinh \relax (x )}{2}+\frac {3 x^{2}}{4}-\frac {3 \left (\cosh ^{2}\relax (x )\right )}{4}+3 x^{2} \sinh \relax (x )-6 x \cosh \relax (x )+6 \sinh \relax (x )+\frac {x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 81, normalized size = 1.45 \[ \frac {1}{4} \, x^{4} + \frac {3}{4} \, x^{2} + \frac {3}{16} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {3}{2} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac {3}{16} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{2} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {3}{8} \, e^{\left (-x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {3}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 48, normalized size = 0.86 \[ \frac {20\,\mathrm {sinh}\relax (x)}{3}+3\,x^2\,\mathrm {sinh}\relax (x)-\frac {3\,{\mathrm {cosh}\relax (x)}^2}{4}+\frac {{\mathrm {cosh}\relax (x)}^2\,\mathrm {sinh}\relax (x)}{3}-6\,x\,\mathrm {cosh}\relax (x)+\frac {3\,x^2}{4}+\frac {x^4}{4}+\frac {3\,x\,\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 85, normalized size = 1.52 \[ \frac {x^{4}}{4} - \frac {3 x^{2} \sinh ^{2}{\relax (x )}}{4} + 3 x^{2} \sinh {\relax (x )} + \frac {3 x^{2} \cosh ^{2}{\relax (x )}}{4} + \frac {3 x \sinh {\relax (x )} \cosh {\relax (x )}}{2} - 6 x \cosh {\relax (x )} - \frac {2 \sinh ^{3}{\relax (x )}}{3} + \sinh {\relax (x )} \cosh ^{2}{\relax (x )} + 6 \sinh {\relax (x )} - \frac {3 \cosh ^{2}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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