Optimal. Leaf size=185 \[ \frac{5 x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{15 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac{15 \text{FresnelC}(b x)^2}{2 \pi ^3 b^7}-\frac{15 x^2}{4 \pi ^3 b^5}+\frac{x^4 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{11 \sin \left (\pi b^2 x^2\right )}{2 \pi ^4 b^7}+\frac{7 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac{x^6}{12 \pi b} \]
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Rubi [A] time = 0.254223, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6463, 6455, 6441, 30, 3380, 2634, 3379, 3296, 2637, 3309} \[ \frac{5 x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{15 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac{15 \text{FresnelC}(b x)^2}{2 \pi ^3 b^7}-\frac{15 x^2}{4 \pi ^3 b^5}+\frac{x^4 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{11 \sin \left (\pi b^2 x^2\right )}{2 \pi ^4 b^7}+\frac{7 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac{x^6}{12 \pi b} \]
Antiderivative was successfully verified.
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Rule 6463
Rule 6455
Rule 6441
Rule 30
Rule 3380
Rule 2634
Rule 3379
Rule 3296
Rule 2637
Rule 3309
Rubi steps
\begin{align*} \int x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{5 \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac{\int x^5 \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{5 x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{15 \int x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac{5 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}+\frac{\operatorname{Subst}\left (\int x^2 \cos ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi }\\ &=\frac{15 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{5 x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{15 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^6 \pi ^3}-\frac{15 \int x \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac{5 \operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}+\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b \pi }+\frac{\operatorname{Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac{x^6}{12 b \pi }+\frac{5 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac{15 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{5 x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{15 \operatorname{Subst}(\int x \, dx,x,C(b x))}{b^7 \pi ^3}-\frac{5 \operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac{15 \operatorname{Subst}\left (\int \cos ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac{\operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac{15 x^2}{4 b^5 \pi ^3}+\frac{x^6}{12 b \pi }+\frac{7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac{15 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac{15 C(b x)^2}{2 b^7 \pi ^3}+\frac{5 x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{5 \sin \left (b^2 \pi x^2\right )}{b^7 \pi ^4}+\frac{x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{\operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}\\ &=-\frac{15 x^2}{4 b^5 \pi ^3}+\frac{x^6}{12 b \pi }+\frac{7 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac{15 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac{15 C(b x)^2}{2 b^7 \pi ^3}+\frac{5 x^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{11 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac{x^4 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.0092096, size = 185, normalized size = 1. \[ \frac{5 x^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{x^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{15 x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac{15 \text{FresnelC}(b x)^2}{2 \pi ^3 b^7}-\frac{15 x^2}{4 \pi ^3 b^5}+\frac{x^4 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{11 \sin \left (\pi b^2 x^2\right )}{2 \pi ^4 b^7}+\frac{7 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac{x^6}{12 \pi b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{x}^{6}{\it FresnelC} \left ( bx \right ) \sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \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^{6}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\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^{6}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 123.898, size = 264, normalized size = 1.43 \begin{align*} \begin{cases} \frac{x^{6} \sin ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} + \frac{x^{6} \cos ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{12 \pi b} - \frac{x^{5} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} + \frac{x^{4} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac{5 x^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac{11 x^{2} \sin ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{3} b^{5}} - \frac{2 x^{2} \cos ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} + \frac{15 x \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} - \frac{11 \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} - \frac{15 C^{2}\left (b x\right )}{2 \pi ^{3} b^{7}} & \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^{6}{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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