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 = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 8
Rule 2635
Rule 3380
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 2.30, size = 27, normalized size = 0.79 \[ \frac {3}{16} \, x^{2} + \frac {1}{16} \, {\left (2 \, \cos \left (x^{2}\right )^{3} + 3 \, \cos \left (x^{2}\right )\right )} \sin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 0.65 \[ \frac {3}{16} \, x^{2} + \frac {1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac {1}{8} \, \sin \left (2 \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 26, normalized size = 0.76 \[ \frac {\left (\cos ^{3}\left (x^{2}\right )+\frac {3 \cos \left (x^{2}\right )}{2}\right ) \sin \left (x^{2}\right )}{8}+\frac {3 x^{2}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 22, normalized size = 0.65 \[ \frac {3}{16} \, x^{2} + \frac {1}{64} \, \sin \left (4 \, x^{2}\right ) + \frac {1}{8} \, \sin \left (2 \, x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 22, normalized size = 0.65 \[ \frac {\sin \left (2\,x^2\right )}{8}+\frac {\sin \left (4\,x^2\right )}{64}+\frac {3\,x^2}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.96, size = 76, normalized size = 2.24 \[ \frac {3 x^{2} \sin ^{4}{\left (x^{2} \right )}}{16} + \frac {3 x^{2} \sin ^{2}{\left (x^{2} \right )} \cos ^{2}{\left (x^{2} \right )}}{8} + \frac {3 x^{2} \cos ^{4}{\left (x^{2} \right )}}{16} + \frac {3 \sin ^{3}{\left (x^{2} \right )} \cos {\left (x^{2} \right )}}{16} + \frac {5 \sin {\left (x^{2} \right )} \cos ^{3}{\left (x^{2} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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