Optimal. Leaf size=150 \[ -\frac{\sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } \sin \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{8 c^{3/2}}-\frac{e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0933062, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3467, 3462, 3448, 3352, 3351} \[ -\frac{\sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } \sin \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{8 c^{3/2}}-\frac{e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3467
Rule 3462
Rule 3448
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int (d+e x) \sin ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{1}{2} (d+e x)-\frac{1}{2} (d+e x) \cos \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{(d+e x)^2}{4 e}-\frac{1}{2} \int (d+e x) \cos \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{(d+e x)^2}{4 e}-\frac{e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{(4 c d-2 b e) \int \cos \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}\\ &=\frac{(d+e x)^2}{4 e}-\frac{e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{\left ((2 c d-b e) \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 ((2 c d-b e) \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{(d+e x)^2}{4 e}-\frac{(2 c d-b e) \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{(2 c d-b e) \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{e \sin \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 0.437429, size = 140, normalized size = 0.93 \[ \frac{\sqrt{\pi } \cos \left (2 a-\frac{b^2}{2 c}\right ) (-(2 c d-b e)) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{\pi } \sqrt{c}}\right )+\sqrt{\pi } \sin \left (2 a-\frac{b^2}{2 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{\pi }}\right )+\sqrt{c} (2 c x (2 d+e x)-e \sin (2 (a+x (b+c x))))}{8 c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 170, normalized size = 1.1 \begin{align*} -{\frac{e\sin \left ( 2\,c{x}^{2}+2\,bx+2\,a \right ) }{8\,c}}+{\frac{be\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}}}}-{\frac{\sqrt{\pi }d}{4} \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 ){\frac{1}{\sqrt{c}}}}+{\frac{dx}{2}}+{\frac{e{x}^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 2.67835, size = 1786, normalized size = 11.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53868, size = 369, normalized size = 2.46 \begin{align*} \frac{2 \, c^{2} e x^{2} - \pi{\left (2 \, c d - b e\right )} \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{\left (2 \, c d - b e\right )} \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 ) + 4 \, c^{2} d x - 2 \, c e \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right ) \sin ^{2}{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.24996, size = 410, normalized size = 2.73 \begin{align*} \frac{1}{4} \, x^{2} e + \frac{1}{2} \, d x + \frac{\sqrt{\pi } d \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 )}}{8 \, \sqrt{c}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )}} + \frac{\sqrt{\pi } d \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 )}}{8 \, \sqrt{c}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )}} - \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}{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 + 1\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}{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 + 1\right )}}{16 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]