Optimal. Leaf size=82 \[ \frac {1}{4} \sqrt {\frac {\pi }{2}} C\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} S\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )+\frac {1}{2} x \sin \left (x^2+x+\frac {1}{4}\right )-\frac {1}{4} \sin \left (x^2+x+\frac {1}{4}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3464, 3445, 3351, 3462, 3446, 3352} \[ \frac {1}{4} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} S\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )+\frac {1}{2} x \sin \left (x^2+x+\frac {1}{4}\right )-\frac {1}{4} \sin \left (x^2+x+\frac {1}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3445
Rule 3446
Rule 3462
Rule 3464
Rubi steps
\begin {align*} \int x^2 \cos \left (\frac {1}{4}+x+x^2\right ) \, dx &=\frac {1}{2} x \sin \left (\frac {1}{4}+x+x^2\right )-\frac {1}{2} \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx-\frac {1}{2} \int \sin \left (\frac {1}{4}+x+x^2\right ) \, dx\\ &=-\frac {1}{4} \sin \left (\frac {1}{4}+x+x^2\right )+\frac {1}{2} x \sin \left (\frac {1}{4}+x+x^2\right )+\frac {1}{4} \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx-\frac {1}{2} \int \sin \left (\frac {1}{4} (1+2 x)^2\right ) \, dx\\ &=-\frac {1}{2} \sqrt {\frac {\pi }{2}} S\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )-\frac {1}{4} \sin \left (\frac {1}{4}+x+x^2\right )+\frac {1}{2} x \sin \left (\frac {1}{4}+x+x^2\right )+\frac {1}{4} \int \cos \left (\frac {1}{4} (1+2 x)^2\right ) \, dx\\ &=\frac {1}{4} \sqrt {\frac {\pi }{2}} C\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} S\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )-\frac {1}{4} \sin \left (\frac {1}{4}+x+x^2\right )+\frac {1}{2} x \sin \left (\frac {1}{4}+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 67, normalized size = 0.82 \[ \frac {1}{8} \left (\sqrt {2 \pi } C\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )-2 \left (\sqrt {2 \pi } S\left (\frac {2 x+1}{\sqrt {2 \pi }}\right )+(1-2 x) \sin \left (x^2+x+\frac {1}{4}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 59, normalized size = 0.72 \[ \frac {1}{8} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, x + 1\right )}}{2 \, \sqrt {\pi }}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, x + 1\right )}}{2 \, \sqrt {\pi }}\right ) + \frac {1}{4} \, {\left (2 \, x - 1\right )} \sin \left (x^{2} + x + \frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.43, size = 75, normalized size = 0.91 \[ -\left (\frac {3}{32} i - \frac {1}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \left (\frac {3}{32} i + \frac {1}{32}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \frac {1}{8} \, {\left (-2 i \, x + i\right )} e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} + \frac {1}{8} \, {\left (2 i \, x - i\right )} e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 59, normalized size = 0.72 \[ \frac {x \sin \left (\frac {1}{4}+x +x^{2}\right )}{2}-\frac {\sin \left (\frac {1}{4}+x +x^{2}\right )}{4}+\frac {\sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{8}-\frac {\sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.76, size = 159, normalized size = 1.94 \[ \frac {x {\left (512 i \, e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} - 512 i \, e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )}\right )} + \sqrt {4 \, x^{2} + 4 \, x + 1} {\left (-\left (32 i - 32\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x^{2} + i \, x + \frac {1}{4} i}\right ) - 1\right )} + \left (32 i + 32\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x^{2} - i \, x - \frac {1}{4} i}\right ) - 1\right )} + \left (128 i + 128\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, i \, x^{2} + i \, x + \frac {1}{4} i\right ) - \left (128 i - 128\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -i \, x^{2} - i \, x - \frac {1}{4} i\right )\right )} + 256 i \, e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} - 256 i \, e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )}}{1024 \, {\left (2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 64, normalized size = 0.78 \[ \frac {x\,\sin \left (x^2+x+\frac {1}{4}\right )}{2}-\frac {\sin \left (x^2+x+\frac {1}{4}\right )}{4}+\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (2\,x+1\right )}{2\,\sqrt {\pi }}\right )}{8}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\left (2\,x+1\right )}{2\,\sqrt {\pi }}\right )}{4} \]
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 \[ \int x^{2} \cos {\left (x^{2} + x + \frac {1}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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