Optimal. Leaf size=46 \[ \frac {x^2}{4}-\frac {1}{8} \sqrt {\pi } \text {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right )+\frac {1}{8} \sin \left (\frac {1}{2}+2 x+2 x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3549, 3543,
3527, 3433} \begin {gather*} -\frac {1}{8} \sqrt {\pi } \text {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x^2}{4}+\frac {1}{8} \sin \left (2 x^2+2 x+\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3527
Rule 3543
Rule 3549
Rubi steps
\begin {align*} \int x \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \left (\frac {x}{2}+\frac {1}{2} x \cos \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac {x^2}{4}+\frac {1}{2} \int x \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x^2}{4}+\frac {1}{8} \sin \left (\frac {1}{2}+2 x+2 x^2\right )-\frac {1}{4} \int \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x^2}{4}+\frac {1}{8} \sin \left (\frac {1}{2}+2 x+2 x^2\right )-\frac {1}{4} \int \cos \left (\frac {1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac {x^2}{4}-\frac {1}{8} \sqrt {\pi } C\left (\frac {1+2 x}{\sqrt {\pi }}\right )+\frac {1}{8} \sin \left (\frac {1}{2}+2 x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 41, normalized size = 0.89 \begin {gather*} \frac {1}{8} \left (2 x^2-\sqrt {\pi } \text {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right )+\sin \left (\frac {1}{2} (1+2 x)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 35, normalized size = 0.76
method | result | size |
default | \(\frac {x^{2}}{4}+\frac {\sin \left (\frac {1}{2}+2 x +2 x^{2}\right )}{8}-\frac {\FresnelC \left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{8}\) | \(35\) |
risch | \(\frac {x^{2}}{4}+\frac {\sqrt {\pi }\, \sqrt {2}\, \left (-1\right )^{\frac {3}{4}} \erf \left (\sqrt {2}\, \left (-1\right )^{\frac {1}{4}} x +\frac {\sqrt {2}\, \left (-1\right )^{\frac {1}{4}}}{2}\right )}{32}-\frac {\sqrt {\pi }\, \erf \left (\sqrt {-2 i}\, x -\frac {i}{\sqrt {-2 i}}\right )}{16 \sqrt {-2 i}}+\frac {\sin \left (\frac {\left (1+2 x \right )^{2}}{2}\right )}{8}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 136, normalized size = 2.96 \begin {gather*} \frac {32 \, x^{3} + 16 \, x^{2} + 8 \, 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 )} + \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 )}\right )} - 4 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} + 4 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )}}{64 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 37, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {1}{4} \, \cos \left (x^{2} + x + \frac {1}{4}\right ) \sin \left (x^{2} + x + \frac {1}{4}\right ) - \frac {1}{8} \, \sqrt {\pi } \operatorname {C}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (37) = 74\).
time = 0.96, size = 121, normalized size = 2.63 \begin {gather*} \frac {x^{2}}{4} - \frac {\sqrt {\pi } x C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{16 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {\pi } x C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right )}{4} + \frac {\sin {\left (2 \left (x + \frac {1}{2}\right )^{2} \right )} \Gamma \left (\frac {1}{4}\right )}{32 \Gamma \left (\frac {5}{4}\right )} - \frac {\sqrt {\pi } C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{32 \Gamma \left (\frac {5}{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 = 0.48, size = 54, normalized size = 1.17 \begin {gather*} \frac {1}{4} \, x^{2} + \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) - \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{2}\right ) - \frac {1}{16} i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} + \frac {1}{16} i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\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 = -0.02 \begin {gather*} \int x\,{\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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