Optimal. Leaf size=34 \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (x^2\right ) \cos ^3\left (x^2\right )+\frac{3}{16} \sin \left (x^2\right ) \cos \left (x^2\right ) \]
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Rubi [A] time = 0.0224351, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3380, 2635, 8} \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (x^2\right ) \cos ^3\left (x^2\right )+\frac{3}{16} \sin \left (x^2\right ) \cos \left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3380
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int x \cos ^4\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cos ^4(x) \, dx,x,x^2\right )\\ &=\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )+\frac{3}{8} \operatorname{Subst}\left (\int \cos ^2(x) \, dx,x,x^2\right )\\ &=\frac{3}{16} \cos \left (x^2\right ) \sin \left (x^2\right )+\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )+\frac{3}{16} \operatorname{Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=\frac{3 x^2}{16}+\frac{3}{16} \cos \left (x^2\right ) \sin \left (x^2\right )+\frac{1}{8} \cos ^3\left (x^2\right ) \sin \left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0167307, size = 28, normalized size = 0.82 \[ \frac{3 x^2}{16}+\frac{1}{8} \sin \left (2 x^2\right )+\frac{1}{64} \sin \left (4 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 26, normalized size = 0.8 \begin{align*}{\frac{\sin \left ({x}^{2} \right ) }{8} \left ( \left ( \cos \left ({x}^{2} \right ) \right ) ^{3}+{\frac{3\,\cos \left ({x}^{2} \right ) }{2}} \right ) }+{\frac{3\,{x}^{2}}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957417, size = 30, normalized size = 0.88 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac{1}{8} \, \sin \left (2 \, x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02743, size = 73, normalized size = 2.15 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{16} \,{\left (2 \, \cos \left (x^{2}\right )^{3} + 3 \, \cos \left (x^{2}\right )\right )} \sin \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.20831, size = 76, normalized size = 2.24 \begin{align*} \frac{3 x^{2} \sin ^{4}{\left (x^{2} \right )}}{16} + \frac{3 x^{2} \sin ^{2}{\left (x^{2} \right )} \cos ^{2}{\left (x^{2} \right )}}{8} + \frac{3 x^{2} \cos ^{4}{\left (x^{2} \right )}}{16} + \frac{3 \sin ^{3}{\left (x^{2} \right )} \cos{\left (x^{2} \right )}}{16} + \frac{5 \sin{\left (x^{2} \right )} \cos ^{3}{\left (x^{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08616, size = 30, normalized size = 0.88 \begin{align*} \frac{3}{16} \, x^{2} + \frac{1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac{1}{8} \, \sin \left (2 \, x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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