Optimal. Leaf size=120 \[ \frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{3 \text{FresnelC}(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}-\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rubi [A] time = 0.118108, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6455, 6463, 6441, 30, 3380, 2634, 3379, 3296, 2637} \[ \frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{3 \text{FresnelC}(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}-\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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Rule 6455
Rule 6463
Rule 6441
Rule 30
Rule 3380
Rule 2634
Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{3 \int x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac{3 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{3 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^4 \pi ^2}-\frac{3 \int x \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac{\operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac{x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{3 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{3 \operatorname{Subst}(\int x \, dx,x,C(b x))}{b^5 \pi ^2}-\frac{\operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac{3 \operatorname{Subst}\left (\int \cos ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac{3 x^2}{4 b^3 \pi ^2}+\frac{x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{3 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac{3 C(b x)^2}{2 b^5 \pi ^2}+\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\sin \left (b^2 \pi x^2\right )}{b^5 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.006659, size = 120, normalized size = 1. \[ \frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{3 \text{FresnelC}(b x)^2}{2 \pi ^2 b^5}-\frac{3 x^2}{4 \pi ^2 b^3}-\frac{\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ){\it FresnelC} \left ( bx \right ) \, dx \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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )\,{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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.6684, size = 151, normalized size = 1.26 \begin{align*} \begin{cases} \frac{x^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac{x^{2} \sin ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{2} b^{3}} - \frac{x^{2} \cos ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac{3 x \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac{2 \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac{3 C^{2}\left (b x\right )}{2 \pi ^{2} b^{5}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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