Optimal. Leaf size=46 \[ \frac {1}{8} \sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x^2}{4}-\frac {1}{8} \sin \left (2 x^2+2 x+\frac {1}{2}\right ) \]
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Rubi [A] time = 0.03, 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 = {3467, 3462, 3446, 3352} \[ \frac {1}{8} \sqrt {\pi } \text {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x^2}{4}-\frac {1}{8} \sin \left (2 x^2+2 x+\frac {1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3446
Rule 3462
Rule 3467
Rubi steps
\begin {align*} \int x \sin ^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 = 0.06, size = 42, normalized size = 0.91 \[ \frac {1}{8} \left (\sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right )+2 x^2-\sin \left (\frac {1}{2} (2 x+1)^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 37, normalized size = 0.80 \[ \frac {1}{4} \, x^{2} - \frac {1}{4} \, \cos \left (x^{2} + x + \frac {1}{4}\right ) \sin \left (x^{2} + x + \frac {1}{4}\right ) + \frac {1}{8} \, \sqrt {\pi } \operatorname {C}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.19, size = 54, normalized size = 1.17 \[ \frac {1}{4} \, x^{2} - \left (\frac {1}{32} i + \frac {1}{32}\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) + \left (\frac {1}{32} i - \frac {1}{32}\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{2}\right ) + \frac {1}{16} i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} - \frac {1}{16} i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 0.76 \[ \frac {x^{2}}{4}-\frac {\sin \left (\frac {1}{2}+2 x +2 x^{2}\right )}{8}+\frac {\FresnelC \left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.54, size = 135, normalized size = 2.93 \[ \frac {65536 \, x^{3} + 32768 \, x^{2} + x {\left (16384 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} - 16384 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )}\right )} + \sqrt {8 \, x^{2} + 8 \, x + 2} {\left (-\left (2048 i - 2048\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {2 i \, x^{2} + 2 i \, x + \frac {1}{2} i}\right ) - 1\right )} + \left (2048 i + 2048\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i}\right ) - 1\right )}\right )} + 8192 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac {1}{2} i\right )} - 8192 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac {1}{2} i\right )}}{131072 \, {\left (2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\sin \left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.91, size = 121, normalized size = 2.63 \[ \frac {x^{2}}{4} - \frac {\sqrt {\pi } x C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right )}{4} + \frac {\sqrt {\pi } x C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{16 \Gamma \left (\frac {5}{4}\right )} - \frac {\sin {\left (2 \left (x + \frac {1}{2}\right )^{2} \right )} \Gamma \left (\frac {1}{4}\right )}{32 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {\pi } C\left (\frac {2 x}{\sqrt {\pi }} + \frac {1}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{32 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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