Optimal. Leaf size=188 \[ \frac{3 \sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sin (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b} \]
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Rubi [A] time = 0.224809, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3403, 3385, 3354, 3352, 3351} \[ \frac{3 \sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sin (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 3385
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int x^2 \sin ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac{3}{4} x^2 \sin \left (a+b x^2\right )-\frac{1}{4} x^2 \sin \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\left (\frac{1}{4} \int x^2 \sin \left (3 a+3 b x^2\right ) \, dx\right )+\frac{3}{4} \int x^2 \sin \left (a+b x^2\right ) \, dx\\ &=-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b}-\frac{\int \cos \left (3 a+3 b x^2\right ) \, dx}{24 b}+\frac{3 \int \cos \left (a+b x^2\right ) \, dx}{8 b}\\ &=-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b}+\frac{(3 \cos (a)) \int \cos \left (b x^2\right ) \, dx}{8 b}-\frac{\cos (3 a) \int \cos \left (3 b x^2\right ) \, dx}{24 b}-\frac{(3 \sin (a)) \int \sin \left (b x^2\right ) \, dx}{8 b}+\frac{\sin (3 a) \int \sin \left (3 b x^2\right ) \, dx}{24 b}\\ &=-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b}+\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) C\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.433139, size = 159, normalized size = 0.85 \[ \frac{27 \sqrt{2 \pi } \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sqrt{6 \pi } \cos (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )-27 \sqrt{2 \pi } \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )+\sqrt{6 \pi } \sin (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )-54 \sqrt{b} x \cos \left (a+b x^2\right )+6 \sqrt{b} x \cos \left (3 \left (a+b x^2\right )\right )}{144 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 132, normalized size = 0.7 \begin{align*} -{\frac{3\,x\cos \left ( b{x}^{2}+a \right ) }{8\,b}}+{\frac{3\,\sqrt{2}\sqrt{\pi }}{16} \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}}+{\frac{x\cos \left ( 3\,b{x}^{2}+3\,a \right ) }{24\,b}}-{\frac{\sqrt{2}\sqrt{\pi }\sqrt{3}}{144} \left ( \cos \left ( 3\,a \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}\sqrt{b}} \right ) -\sin \left ( 3\,a \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.24511, size = 695, normalized size = 3.7 \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 = 2.46357, size = 447, normalized size = 2.38 \begin{align*} \frac{24 \, b x \cos \left (b x^{2} + a\right )^{3} - \sqrt{6} \pi \sqrt{\frac{b}{\pi }} \cos \left (3 \, a\right ) \operatorname{C}\left (\sqrt{6} x \sqrt{\frac{b}{\pi }}\right ) + 27 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) + \sqrt{6} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{6} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (3 \, a\right ) - 27 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) - 72 \, b x \cos \left (b x^{2} + a\right )}{144 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.3889, size = 439, normalized size = 2.34 \begin{align*} - \frac{3 b^{\frac{3}{2}} x^{5} \sqrt{\frac{1}{b}} \cos{\left (a \right )} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle |{- \frac{b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{9}{4}\right )} + \frac{3 b^{\frac{3}{2}} x^{5} \sqrt{\frac{1}{b}} \cos{\left (3 a \right )} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle |{- \frac{9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{9}{4}\right )} - \frac{3 \sqrt{b} x^{3} \sqrt{\frac{1}{b}} \sin{\left (a \right )} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{b} x^{3} \sqrt{\frac{1}{b}} \sin{\left (3 a \right )} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} + \frac{3 \sqrt{2} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \sin{\left (a \right )} C\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right )}{8} - \frac{\sqrt{6} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \sin{\left (3 a \right )} C\left (\frac{\sqrt{6} \sqrt{b} x}{\sqrt{\pi }}\right )}{24} + \frac{3 \sqrt{2} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \cos{\left (a \right )} S\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right )}{8} - \frac{\sqrt{6} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \cos{\left (3 a \right )} S\left (\frac{\sqrt{6} \sqrt{b} x}{\sqrt{\pi }}\right )}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.14382, size = 350, normalized size = 1.86 \begin{align*} \frac{x e^{\left (3 i \, b x^{2} + 3 i \, a\right )}}{48 \, b} - \frac{3 \, x e^{\left (i \, b x^{2} + i \, a\right )}}{16 \, b} - \frac{3 \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{16 \, b} + \frac{x e^{\left (-3 i \, b x^{2} - 3 i \, a\right )}}{48 \, b} + \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{6} \sqrt{b} x{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{288 \, b^{\frac{3}{2}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}} - \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{32 \, b{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} - \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} x{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{32 \, b{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} + \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{6} \sqrt{b} x{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{288 \, b^{\frac{3}{2}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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