Optimal. Leaf size=37 \[ x^5 \sinh (x)+20 x^3 \sinh (x)-5 x^4 \cosh (x)-60 x^2 \cosh (x)+120 x \sinh (x)-120 \cosh (x) \]
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Rubi [A] time = 0.0764606, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3296, 2638} \[ x^5 \sinh (x)+20 x^3 \sinh (x)-5 x^4 \cosh (x)-60 x^2 \cosh (x)+120 x \sinh (x)-120 \cosh (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^5 \cosh (x) \, dx &=x^5 \sinh (x)-5 \int x^4 \sinh (x) \, dx\\ &=-5 x^4 \cosh (x)+x^5 \sinh (x)+20 \int x^3 \cosh (x) \, dx\\ &=-5 x^4 \cosh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)-60 \int x^2 \sinh (x) \, dx\\ &=-60 x^2 \cosh (x)-5 x^4 \cosh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)+120 \int x \cosh (x) \, dx\\ &=-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)-120 \int \sinh (x) \, dx\\ &=-120 \cosh (x)-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0158831, size = 29, normalized size = 0.78 \[ x \left (x^4+20 x^2+120\right ) \sinh (x)-5 \left (x^4+12 x^2+24\right ) \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 38, normalized size = 1. \begin{align*} -120\,\cosh \left ( x \right ) -60\,{x}^{2}\cosh \left ( x \right ) -5\,{x}^{4}\cosh \left ( x \right ) +120\,x\sinh \left ( x \right ) +20\,{x}^{3}\sinh \left ( x \right ) +{x}^{5}\sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969414, size = 100, normalized size = 2.7 \begin{align*} \frac{1}{6} \, x^{6} \cosh \left (x\right ) - \frac{1}{12} \,{\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} - \frac{1}{12} \,{\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82199, size = 88, normalized size = 2.38 \begin{align*} -5 \,{\left (x^{4} + 12 \, x^{2} + 24\right )} \cosh \left (x\right ) +{\left (x^{5} + 20 \, x^{3} + 120 \, 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 = 2.09731, size = 42, normalized size = 1.14 \begin{align*} x^{5} \sinh{\left (x \right )} - 5 x^{4} \cosh{\left (x \right )} + 20 x^{3} \sinh{\left (x \right )} - 60 x^{2} \cosh{\left (x \right )} + 120 x \sinh{\left (x \right )} - 120 \cosh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04601, size = 77, normalized size = 2.08 \begin{align*} -\frac{1}{2} \,{\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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