Optimal. Leaf size=123 \[ -\frac{\sqrt{\frac{\pi }{2}} b \cos \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b \sin \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}+\frac{\sin \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0408168, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3462, 3448, 3352, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} b \cos \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} b \sin \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}+\frac{\sin \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 3462
Rule 3448
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int x \cos \left (a+b x+c x^2\right ) \, dx &=\frac{\sin \left (a+b x+c x^2\right )}{2 c}-\frac{b \int \cos \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac{\sin \left (a+b x+c x^2\right )}{2 c}-\frac{\left (b \cos \left (a-\frac{b^2}{4 c}\right )\right ) \int \cos \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}+\frac{\left (b \sin \left (a-\frac{b^2}{4 c}\right )\right ) \int \sin \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}\\ &=-\frac{b \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}+\frac{b \sqrt{\frac{\pi }{2}} S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right ) \sin \left (a-\frac{b^2}{4 c}\right )}{2 c^{3/2}}+\frac{\sin \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.32614, size = 112, normalized size = 0.91 \[ \frac{\sqrt{2 \pi } (-b) \cos \left (a-\frac{b^2}{4 c}\right ) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )+\sqrt{2 \pi } b \sin \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )+2 \sqrt{c} \sin (a+x (b+c x))}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 99, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( c{x}^{2}+bx+a \right ) }{2\,c}}-{\frac{\sqrt{2}b\sqrt{\pi }}{4} \left ( \cos \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\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 = 2.06807, size = 1307, normalized size = 10.63 \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 = 1.49369, size = 325, normalized size = 2.64 \begin{align*} -\frac{\sqrt{2} \pi b \sqrt{\frac{c}{\pi }} \cos \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) - \sqrt{2} \pi b \sqrt{\frac{c}{\pi }} \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, c \sin \left (c x^{2} + b x + a\right )}{4 \, 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 \cos{\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 = 1.24136, size = 244, normalized size = 1.98 \begin{align*} \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} - 2 i \, e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{8 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} + 2 i \, e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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