Optimal. Leaf size=71 \[ -\frac {1}{2} x \cos \left (x^2\right )+\frac {1}{6} x \cos ^3\left (x^2\right )+\frac {3}{8} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} x\right )-\frac {1}{24} \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3484, 3466,
3433} \begin {gather*} \frac {3}{8} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} x\right )-\frac {1}{24} \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} x\right )-\frac {3}{8} x \cos \left (x^2\right )+\frac {1}{24} x \cos \left (3 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3466
Rule 3484
Rubi steps
\begin {align*} \int x^2 \sin ^3\left (x^2\right ) \, dx &=\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\\ &=-\left (\frac {1}{4} \int x^2 \sin \left (3 x^2\right ) \, dx\right )+\frac {3}{4} \int x^2 \sin \left (x^2\right ) \, dx\\ &=-\frac {3}{8} x \cos \left (x^2\right )+\frac {1}{24} x \cos \left (3 x^2\right )-\frac {1}{24} \int \cos \left (3 x^2\right ) \, dx+\frac {3}{8} \int \cos \left (x^2\right ) \, dx\\ &=-\frac {3}{8} x \cos \left (x^2\right )+\frac {1}{24} x \cos \left (3 x^2\right )+\frac {3}{8} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} x\right )-\frac {1}{24} \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} x\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 63, normalized size = 0.89 \begin {gather*} \frac {1}{144} \left (6 x \left (-9 \cos \left (x^2\right )+\cos \left (3 x^2\right )\right )+27 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} x\right )-\sqrt {6 \pi } C\left (\sqrt {\frac {6}{\pi }} x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 58, normalized size = 0.82
method | result | size |
default | \(-\frac {3 x \cos \left (x^{2}\right )}{8}+\frac {3 \FresnelC \left (\frac {x \sqrt {2}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{16}+\frac {x \cos \left (3 x^{2}\right )}{24}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, x}{\sqrt {\pi }}\right )}{144}\) | \(58\) |
risch | \(-\frac {\sqrt {\pi }\, \erf \left (\sqrt {-3 i}\, x \right )}{96 \sqrt {-3 i}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \sqrt {3}\, \erf \left (\sqrt {3}\, \left (-1\right )^{\frac {1}{4}} x \right )}{288}-\frac {3 \left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \erf \left (\left (-1\right )^{\frac {1}{4}} x \right )}{32}+\frac {3 \sqrt {\pi }\, \erf \left (\sqrt {-i}\, x \right )}{32 \sqrt {-i}}-\frac {3 x \cos \left (x^{2}\right )}{8}+\frac {x \cos \left (3 x^{2}\right )}{24}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 97, normalized size = 1.37 \begin {gather*} \frac {1}{24} \, x \cos \left (3 \, x^{2}\right ) - \frac {3}{8} \, x \cos \left (x^{2}\right ) + \frac {1}{1152} \, \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 51, normalized size = 0.72 \begin {gather*} \frac {1}{6} \, x \cos \left (x^{2}\right )^{3} - \frac {1}{2} \, x \cos \left (x^{2}\right ) - \frac {1}{144} \, \sqrt {6} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) + \frac {3}{16} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} x}{\sqrt {\pi }}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.15, size = 116, normalized size = 1.63 \begin {gather*} - \frac {15 x \cos {\left (x^{2} \right )} \Gamma \left (\frac {5}{4}\right )}{32 \Gamma \left (\frac {9}{4}\right )} + \frac {5 x \cos {\left (3 x^{2} \right )} \Gamma \left (\frac {5}{4}\right )}{96 \Gamma \left (\frac {9}{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 {9}{4}\right )} - \frac {5 \sqrt {6} \sqrt {\pi } C\left (\frac {\sqrt {6} x}{\sqrt {\pi }}\right ) \Gamma \left (\frac {5}{4}\right )}{576 \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.71, size = 97, normalized size = 1.37 \begin {gather*} \left (\frac {1}{576} i + \frac {1}{576}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} x\right ) - \left (\frac {1}{576} i - \frac {1}{576}\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} x\right ) - \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} x\right ) + \frac {1}{48} \, x e^{\left (3 i \, x^{2}\right )} - \frac {3}{16} \, x e^{\left (i \, x^{2}\right )} - \frac {3}{16} \, x e^{\left (-i \, x^{2}\right )} + \frac {1}{48} \, x e^{\left (-3 i \, x^{2}\right )} \end {gather*}
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 \begin {gather*} \int x^2\,{\sin \left (x^2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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