Optimal. Leaf size=84 \[ \frac {1}{4} x \cos \left (x^2\right )-\frac {1}{12} x \cos ^3\left (x^2\right )-\frac {3}{16} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} x\right )+\frac {1}{48} \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} x\right )+\frac {1}{6} x^3 \sin ^3\left (x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, 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 = {3524, 3484,
3466, 3433} \begin {gather*} -\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 {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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3433
Rule 3466
Rule 3484
Rule 3524
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 75, normalized size = 0.89 \begin {gather*} \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 (9 \cos \left (x^2\right )-\cos \left (3 x^2\right )+8 x^2 \sin ^3\left (x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 78, normalized size = 0.93
method | result | size |
default | \(\frac {x^{3} \sin \left (x^{2}\right )}{8}+\frac {3 x \cos \left (x^{2}\right )}{16}-\frac {3 \FresnelC \left (\frac {x \sqrt {2}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{32}-\frac {x^{3} \sin \left (3 x^{2}\right )}{24}-\frac {x \cos \left (3 x^{2}\right )}{48}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, x}{\sqrt {\pi }}\right )}{288}\) | \(78\) |
risch | \(-\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \sqrt {3}\, \erf \left (\sqrt {3}\, \left (-1\right )^{\frac {1}{4}} x \right )}{576}+\frac {3 \left (-1\right )^{\frac {3}{4}} \sqrt {\pi }\, \erf \left (\left (-1\right )^{\frac {1}{4}} x \right )}{64}+\frac {\sqrt {\pi }\, \erf \left (\sqrt {-3 i}\, x \right )}{192 \sqrt {-3 i}}-\frac {3 \sqrt {\pi }\, \erf \left (\sqrt {-i}\, x \right )}{64 \sqrt {-i}}+\frac {3 x \cos \left (x^{2}\right )}{16}+\frac {x^{3} \sin \left (x^{2}\right )}{8}-\frac {x \cos \left (3 x^{2}\right )}{48}-\frac {x^{3} \sin \left (3 x^{2}\right )}{24}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 117, normalized size = 1.39 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 73, normalized size = 0.87 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (82) = 164\).
time = 2.20, size = 291, normalized size = 3.46 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 3.08, size = 125, normalized size = 1.49 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\cos \left (x^2\right )\,{\sin \left (x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________