Optimal. Leaf size=85 \[ \frac {x^3}{6}+\frac {1}{16} \sqrt {\pi } \text {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\pi } S\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\frac {1}{16} \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} x \sin \left (\frac {1}{2}+2 x+2 x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3549, 3545,
3543, 3527, 3433, 3526, 3432} \begin {gather*} \frac {1}{16} \sqrt {\pi } \text {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\pi } S\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x^3}{6}+\frac {1}{8} x \sin \left (2 x^2+2 x+\frac {1}{2}\right )-\frac {1}{16} \sin \left (2 x^2+2 x+\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3526
Rule 3527
Rule 3543
Rule 3545
Rule 3549
Rubi steps
\begin {align*} \int x^2 \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \left (\frac {x^2}{2}+\frac {1}{2} x^2 \cos \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac {x^3}{6}+\frac {1}{2} \int x^2 \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x^3}{6}+\frac {1}{8} x \sin \left (\frac {1}{2}+2 x+2 x^2\right )-\frac {1}{8} \int \sin \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx-\frac {1}{4} \int x \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x^3}{6}-\frac {1}{16} \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} x \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} \int \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx-\frac {1}{8} \int \sin \left (\frac {1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac {x^3}{6}-\frac {1}{16} \sqrt {\pi } S\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\frac {1}{16} \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} x \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} \int \cos \left (\frac {1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac {x^3}{6}+\frac {1}{16} \sqrt {\pi } C\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\pi } S\left (\frac {1+2 x}{\sqrt {\pi }}\right )-\frac {1}{16} \sin \left (\frac {1}{2}+2 x+2 x^2\right )+\frac {1}{8} x \sin \left (\frac {1}{2}+2 x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {1}{48} \left (8 x^3+3 \sqrt {\pi } \text {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right )-3 \sqrt {\pi } S\left (\frac {1+2 x}{\sqrt {\pi }}\right )-3 \sin \left (\frac {1}{2} (1+2 x)^2\right )+6 x \sin \left (\frac {1}{2} (1+2 x)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 64, normalized size = 0.75
method | result | size |
default | \(\frac {x^{3}}{6}-\frac {\sin \left (\frac {1}{2}+2 x +2 x^{2}\right )}{16}+\frac {x \sin \left (\frac {1}{2}+2 x +2 x^{2}\right )}{8}+\frac {\FresnelC \left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{16}-\frac {\mathrm {S}\left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{16}\) | \(64\) |
risch | \(\frac {x^{3}}{6}-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (-1\right )^{\frac {3}{4}} \erf \left (\sqrt {2}\, \left (-1\right )^{\frac {1}{4}} x +\frac {\sqrt {2}\, \left (-1\right )^{\frac {1}{4}}}{2}\right )}{64}-\frac {\left (-1\right )^{\frac {1}{4}} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, \left (-1\right )^{\frac {1}{4}} x +\frac {\sqrt {2}\, \left (-1\right )^{\frac {1}{4}}}{2}\right )}{64}+\frac {\sqrt {\pi }\, \erf \left (\sqrt {-2 i}\, x -\frac {i}{\sqrt {-2 i}}\right )}{32 \sqrt {-2 i}}+\frac {i \sqrt {\pi }\, \erf \left (\sqrt {-2 i}\, x -\frac {i}{\sqrt {-2 i}}\right )}{32 \sqrt {-2 i}}+2 i \left (-\frac {1}{16} i x +\frac {1}{32} i\right ) \sin \left (\frac {\left (1+2 x \right )^{2}}{2}\right )\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.60, size = 171, normalized size = 2.01 \begin {gather*} \frac {128 \, x^{4} + 64 \, x^{3} - 48 \, x {\left (-i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} + i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )}\right )} - 3 \, \sqrt {8 \, x^{2} + 8 \, x + 2} {\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {2 i \, x^{2} + 2 i \, x + \frac {1}{2} i}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i}\right ) - 1\right )} - \left (2 i + 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, 2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right ) + \left (2 i - 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )\right )} + 24 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} - 24 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )}}{384 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 57, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, x^{3} + \frac {1}{8} \, {\left (2 \, x - 1\right )} \cos \left (x^{2} + x + \frac {1}{4}\right ) \sin \left (x^{2} + x + \frac {1}{4}\right ) + \frac {1}{16} \, \sqrt {\pi } \operatorname {C}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) - \frac {1}{16} \, \sqrt {\pi } \operatorname {S}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \cos ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}\, dx \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 = 0.45, size = 64, normalized size = 0.75 \begin {gather*} \frac {1}{6} \, x^{3} - \frac {1}{32} \, {\left (2 i \, x - i\right )} e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} - \frac {1}{32} \, {\left (-2 i \, x + i\right )} e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )} - \frac {1}{32} i \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) + \frac {1}{32} i \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{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\,{\cos \left (x^2+x+\frac {1}{4}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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