Optimal. Leaf size=84 \[ -\frac {3}{16} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} x\right )+\frac {1}{48} \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} x\right )-\frac {1}{12} x \cos ^3\left (x^2\right )+\frac {1}{4} x \cos \left (x^2\right )+\frac {1}{6} x^3 \sin ^3\left (x^2\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3443, 3403, 3385, 3352} \[ -\frac {3}{16} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} x\right )+\frac {1}{48} \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} x\right )+\frac {1}{6} x^3 \sin ^3\left (x^2\right )+\frac {3}{16} x \cos \left (x^2\right )-\frac {1}{48} x \cos \left (3 x^2\right ) \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3385
Rule 3403
Rule 3443
Rubi steps
\begin {align*} \int x^4 \cos \left (x^2\right ) \sin ^2\left (x^2\right ) \, dx &=\frac {1}{6} x^3 \sin ^3\left (x^2\right )-\frac {1}{2} \int x^2 \sin ^3\left (x^2\right ) \, dx\\ &=\frac {1}{6} x^3 \sin ^3\left (x^2\right )-\frac {1}{2} \int \left (\frac {3}{4} x^2 \sin \left (x^2\right )-\frac {1}{4} x^2 \sin \left (3 x^2\right )\right ) \, dx\\ &=\frac {1}{6} x^3 \sin ^3\left (x^2\right )+\frac {1}{8} \int x^2 \sin \left (3 x^2\right ) \, dx-\frac {3}{8} \int x^2 \sin \left (x^2\right ) \, dx\\ &=\frac {3}{16} x \cos \left (x^2\right )-\frac {1}{48} x \cos \left (3 x^2\right )+\frac {1}{6} x^3 \sin ^3\left (x^2\right )+\frac {1}{48} \int \cos \left (3 x^2\right ) \, dx-\frac {3}{16} \int \cos \left (x^2\right ) \, dx\\ &=\frac {3}{16} x \cos \left (x^2\right )-\frac {1}{48} x \cos \left (3 x^2\right )-\frac {3}{16} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} x\right )+\frac {1}{48} \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} x\right )+\frac {1}{6} x^3 \sin ^3\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 75, normalized size = 0.89 \[ \frac {1}{288} \left (-27 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} x\right )+\sqrt {6 \pi } C\left (\sqrt {\frac {6}{\pi }} x\right )+6 x \left (8 x^2 \sin ^3\left (x^2\right )+9 \cos \left (x^2\right )-\cos \left (3 x^2\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 73, normalized size = 0.87 \[ -\frac {1}{12} \, x \cos \left (x^{2}\right )^{3} + \frac {1}{4} \, x \cos \left (x^{2}\right ) + \frac {1}{288} \, \sqrt {6} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) - \frac {3}{32} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) - \frac {1}{6} \, {\left (x^{3} \cos \left (x^{2}\right )^{2} - x^{3}\right )} \sin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.74, size = 125, normalized size = 1.49 \[ -\left (\frac {1}{1152} i + \frac {1}{1152}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} x\right ) + \left (\frac {1}{1152} i - \frac {1}{1152}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} x\right ) + \left (\frac {3}{128} i + \frac {3}{128}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} x\right ) - \left (\frac {3}{128} i - \frac {3}{128}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} x\right ) - \frac {1}{96} \, {\left (-2 i \, x^{3} + x\right )} e^{\left (3 i \, x^{2}\right )} - \frac {1}{32} \, {\left (2 i \, x^{3} - 3 \, x\right )} e^{\left (i \, x^{2}\right )} - \frac {1}{32} \, {\left (-2 i \, x^{3} - 3 \, x\right )} e^{\left (-i \, x^{2}\right )} - \frac {1}{96} \, {\left (2 i \, x^{3} + x\right )} e^{\left (-3 i \, x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 78, normalized size = 0.93 \[ \frac {x^{3} \sin \left (x^{2}\right )}{8}+\frac {3 x \cos \left (x^{2}\right )}{16}-\frac {3 \FresnelC \left (\frac {x \sqrt {2}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{32}-\frac {x^{3} \sin \left (3 x^{2}\right )}{24}-\frac {x \cos \left (3 x^{2}\right )}{48}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, x}{\sqrt {\pi }}\right )}{288} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.81, size = 117, normalized size = 1.39 \[ -\frac {1}{24} \, x^{3} \sin \left (3 \, x^{2}\right ) + \frac {1}{8} \, x^{3} \sin \left (x^{2}\right ) - \frac {1}{48} \, x \cos \left (3 \, x^{2}\right ) + \frac {3}{16} \, x \cos \left (x^{2}\right ) - \frac {1}{2304} \, \sqrt {\pi } {\left (\left (2 i - 2\right ) \, \sqrt {3} \sqrt {2} \operatorname {erf}\left (\sqrt {3 i} x\right ) - \left (2 i + 2\right ) \, \sqrt {3} \sqrt {2} \operatorname {erf}\left (\sqrt {-3 i} x\right ) - \left (27 i - 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} x\right ) - \left (27 i + 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \left (27 i + 27\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} x\right ) - \left (27 i - 27\right ) \, \sqrt {2} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\cos \left (x^2\right )\,{\sin \left (x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.91, size = 291, normalized size = 3.46 \[ - \frac {9 x^{5} \Gamma \left (- \frac {9}{4}\right )}{40 \Gamma \left (- \frac {5}{4}\right )} + \frac {9 x^{3} \sin {\left (x^{2} \right )} \Gamma \left (- \frac {9}{4}\right )}{32 \Gamma \left (- \frac {5}{4}\right )} - \frac {5 x^{3} \sin {\left (x^{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{16 \Gamma \left (- \frac {1}{4}\right )} + \frac {3 x^{3} \sin {\left (3 x^{2} \right )} \Gamma \left (- \frac {9}{4}\right )}{32 \Gamma \left (- \frac {5}{4}\right )} + \frac {27 x \cos {\left (x^{2} \right )} \Gamma \left (- \frac {9}{4}\right )}{64 \Gamma \left (- \frac {5}{4}\right )} - \frac {15 x \cos {\left (x^{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{32 \Gamma \left (- \frac {1}{4}\right )} + \frac {3 x \cos {\left (3 x^{2} \right )} \Gamma \left (- \frac {9}{4}\right )}{64 \Gamma \left (- \frac {5}{4}\right )} + \frac {15 \sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {5}{4}\right )}{64 \Gamma \left (- \frac {1}{4}\right )} - \frac {27 \sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {9}{4}\right )}{128 \Gamma \left (- \frac {5}{4}\right )} - \frac {\sqrt {6} \sqrt {\pi } C\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {9}{4}\right )}{128 \Gamma \left (- \frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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