Optimal. Leaf size=27 \[ \frac {x}{2}-\frac {1}{4} \sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3449, 3446, 3352} \[ \frac {x}{2}-\frac {1}{4} \sqrt {\pi } \text {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right ) \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3446
Rule 3449
Rubi steps
\begin {align*} \int \sin ^2\left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} \int \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} \int \cos \left (\frac {1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{4} \sqrt {\pi } C\left (\frac {1+2 x}{\sqrt {\pi }}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {1}{4} \left (2 x-\sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 0.70 \[ -\frac {1}{4} \, \sqrt {\pi } \operatorname {C}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 26, normalized size = 0.96 \[ \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{2}\right ) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.74 \[ \frac {x}{2}-\frac {\FresnelC \left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.43, size = 34, normalized size = 1.26 \[ \frac {1}{16} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {2 i}}\right ) + \left (i + 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {-2 i}}\right )\right )} + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\sin \left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 22, normalized size = 0.81 \[ \frac {x}{2} - \frac {\sqrt {\pi } C\left (\frac {4 x + 2}{2 \sqrt {\pi }}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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