Optimal. Leaf size=126 \[ \frac{\sqrt{\pi } b \cos \left (2 a-\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b \sin \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{8 c^{3/2}}-\frac{\sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{x^2}{4} \]
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Rubi [A] time = 0.0728105, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3467, 3462, 3448, 3352, 3351} \[ \frac{\sqrt{\pi } b \cos \left (2 a-\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } b \sin \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{8 c^{3/2}}-\frac{\sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{x^2}{4} \]
Antiderivative was successfully verified.
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Rule 3467
Rule 3462
Rule 3448
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int x \sin ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{x}{2}-\frac{1}{2} x \cos \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{x^2}{4}-\frac{1}{2} \int x \cos \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{x^2}{4}-\frac{\sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{b \int \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{x^2}{4}-\frac{\sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left (b \cos \left (2 a-\frac{b^2}{2 c}\right )\right ) \int \cos \left (\frac{(2 b+4 c x)^2}{8 c}\right ) \, dx}{4 c}-\frac{\left (b \sin \left (2 a-\frac{b^2}{2 c}\right )\right ) \int \sin \left (\frac{(2 b+4 c x)^2}{8 c}\right ) \, dx}{4 c}\\ &=\frac{x^2}{4}+\frac{b \sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{8 c^{3/2}}-\frac{b \sqrt{\pi } S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{b^2}{2 c}\right )}{8 c^{3/2}}-\frac{\sin \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 0.280254, size = 118, normalized size = 0.94 \[ \frac{\sqrt{\pi } b \cos \left (2 a-\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )-\sqrt{\pi } b \sin \left (2 a-\frac{b^2}{2 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )+\sqrt{c} \left (2 c x^2-\sin (2 (a+x (b+c x)))\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 95, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{\sin \left ( 2\,c{x}^{2}+2\,bx+2\,a \right ) }{8\,c}}+{\frac{b\sqrt{\pi }}{8} \left ( \cos \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{-4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx+b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.1199, size = 1372, normalized size = 10.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63334, size = 315, normalized size = 2.5 \begin{align*} \frac{\pi b \sqrt{\frac{c}{\pi }} \cos \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right ) \operatorname{C}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{c}\right ) - \pi b \sqrt{\frac{c}{\pi }} \operatorname{S}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{c}\right ) \sin \left (-\frac{b^{2} - 4 \, a c}{2 \, c}\right ) + 2 \, c^{2} x^{2} - 2 \, c \cos \left (c x^{2} + b x + a\right ) \sin \left (c x^{2} + b x + a\right )}{8 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin ^{2}{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.24837, size = 230, normalized size = 1.83 \begin{align*} \frac{1}{4} \, x^{2} - \frac{\frac{\sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac{i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{\sqrt{c}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )}} - i \, e^{\left (2 i \, c x^{2} + 2 i \, b x + 2 i \, a\right )}}{16 \, c} - \frac{\frac{\sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac{-i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{\sqrt{c}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )}} + i \, e^{\left (-2 i \, c x^{2} - 2 i \, b x - 2 i \, a\right )}}{16 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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