Optimal. Leaf size=177 \[ -\frac{2 x^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac{16 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac{8 x^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}-\frac{43 S\left (\sqrt{2} b x\right )}{20 \sqrt{2} \pi ^3 b^5}+\frac{4 x^3}{15 \pi ^2 b^2}+\frac{11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}-\frac{x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}+\frac{1}{5} x^5 \text{FresnelC}(b x)^2 \]
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Rubi [A] time = 0.187988, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {6431, 6455, 6463, 6453, 3351, 3392, 30, 3386, 3385} \[ -\frac{2 x^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac{16 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac{8 x^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}-\frac{43 S\left (\sqrt{2} b x\right )}{20 \sqrt{2} \pi ^3 b^5}+\frac{4 x^3}{15 \pi ^2 b^2}+\frac{11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}-\frac{x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}+\frac{1}{5} x^5 \text{FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6431
Rule 6455
Rule 6463
Rule 6453
Rule 3351
Rule 3392
Rule 30
Rule 3386
Rule 3385
Rubi steps
\begin{align*} \int x^4 C(b x)^2 \, dx &=\frac{1}{5} x^5 C(b x)^2-\frac{1}{5} (2 b) \int x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac{1}{5} x^5 C(b x)^2-\frac{2 x^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac{\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{5 \pi }+\frac{8 \int x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{5 b \pi }\\ &=-\frac{x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac{8 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac{1}{5} x^5 C(b x)^2-\frac{2 x^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac{16 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{5 b^3 \pi ^2}+\frac{3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{10 b^2 \pi ^2}+\frac{8 \int x^2 \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=-\frac{x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac{8 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac{1}{5} x^5 C(b x)^2+\frac{16 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac{2 x^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac{3 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}-\frac{3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{20 b^4 \pi ^3}-\frac{8 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}+\frac{4 \int x^2 \, dx}{5 b^2 \pi ^2}+\frac{4 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=\frac{4 x^3}{15 b^2 \pi ^2}-\frac{x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac{8 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac{1}{5} x^5 C(b x)^2-\frac{3 S\left (\sqrt{2} b x\right )}{20 \sqrt{2} b^5 \pi ^3}-\frac{4 \sqrt{2} S\left (\sqrt{2} b x\right )}{5 b^5 \pi ^3}+\frac{16 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac{2 x^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac{11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}-\frac{2 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}\\ &=\frac{4 x^3}{15 b^2 \pi ^2}-\frac{x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac{8 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac{1}{5} x^5 C(b x)^2-\frac{3 S\left (\sqrt{2} b x\right )}{20 \sqrt{2} b^5 \pi ^3}-\frac{\sqrt{2} S\left (\sqrt{2} b x\right )}{b^5 \pi ^3}+\frac{16 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac{2 x^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac{11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.133759, size = 137, normalized size = 0.77 \[ \frac{24 \pi ^3 b^5 x^5 \text{FresnelC}(b x)^2-48 \text{FresnelC}(b x) \left (\left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+4 \pi b^2 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )\right )+32 \pi b^3 x^3+66 b x \sin \left (\pi b^2 x^2\right )-12 \pi b^3 x^3 \cos \left (\pi b^2 x^2\right )-129 \sqrt{2} S\left (\sqrt{2} b x\right )}{120 \pi ^3 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 209, normalized size = 1.2 \begin{align*}{\frac{1}{{b}^{5}} \left ({\frac{{b}^{5}{x}^{5} \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}}{5}}-2\,{\it FresnelC} \left ( bx \right ) \left ( 1/5\,{\frac{{x}^{4}{b}^{4}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-4/5\,{\frac{1}{\pi } \left ( -{\frac{{b}^{2}{x}^{2}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2\,{\frac{\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) } \right ) +{\frac{4\,{x}^{3}{b}^{3}}{15\,{\pi }^{2}}}+{\frac{4}{5\,{\pi }^{2}} \left ({\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) }+{\frac{1}{5\,{\pi }^{3}} \left ( -{\frac{\pi \,{b}^{3}{x}^{3}\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}+{\frac{3\,\pi }{2} \left ({\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) }-4\,\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\rm fresnelc}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4}{\rm fresnelc}\left (b x\right )^{2}, x\right ) \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^{4} C^{2}\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\rm fresnelc}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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