Optimal. Leaf size=140 \[ -\frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}-\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{3 \text{FresnelC}(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}+\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 \text{FresnelC}(b x)^2 \]
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Rubi [A] time = 0.155816, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6431, 6455, 6463, 6441, 30, 3380, 2634, 3379, 3296, 2637} \[ -\frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}-\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{3 \text{FresnelC}(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}+\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 \text{FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6431
Rule 6455
Rule 6463
Rule 6441
Rule 30
Rule 3380
Rule 2634
Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 C(b x)^2 \, dx &=\frac{1}{4} x^4 C(b x)^2-\frac{1}{2} b \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac{1}{4} x^4 C(b x)^2-\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac{\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{4 \pi }+\frac{3 \int x^2 C(b x) \sin \left (\frac{1}{2} 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)}{2 b^3 \pi ^2}+\frac{1}{4} x^4 C(b x)^2-\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac{3 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{2 b^3 \pi ^2}+\frac{3 \int x \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{2 b^2 \pi ^2}+\frac{\operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 \pi }\\ &=-\frac{x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac{3 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{2 b^3 \pi ^2}+\frac{1}{4} x^4 C(b x)^2-\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac{3 \operatorname{Subst}(\int x \, dx,x,C(b x))}{2 b^4 \pi ^2}+\frac{\operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}+\frac{3 \operatorname{Subst}\left (\int \cos ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^2 \pi ^2}\\ &=\frac{3 x^2}{8 b^2 \pi ^2}-\frac{x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac{3 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{2 b^3 \pi ^2}+\frac{3 C(b x)^2}{4 b^4 \pi ^2}+\frac{1}{4} x^4 C(b x)^2-\frac{x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac{\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.0087466, size = 140, normalized size = 1. \[ -\frac{x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}-\frac{3 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{3 \text{FresnelC}(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}+\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 \text{FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}\, 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^{3}{\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^{3}{\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^{3} 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^{3}{\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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