Optimal. Leaf size=109 \[ \frac {x^6 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac {48 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}-\frac {24 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac {6 x^4 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {1}{7} x^7 S(b x) \]
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Rubi [A] time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3379, 3296, 2637} \[ -\frac {6 x^4 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {48 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac {x^6 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}-\frac {24 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac {1}{7} x^7 S(b x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3379
Rule 6426
Rubi steps
\begin {align*} \int x^6 S(b x) \, dx &=\frac {1}{7} x^7 S(b x)-\frac {1}{7} b \int x^7 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{7} x^7 S(b x)-\frac {1}{14} b \operatorname {Subst}\left (\int x^3 \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )\\ &=\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {1}{7} x^7 S(b x)-\frac {3 \operatorname {Subst}\left (\int x^2 \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b \pi }\\ &=\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {1}{7} x^7 S(b x)-\frac {6 x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {12 \operatorname {Subst}\left (\int x \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^3 \pi ^2}\\ &=-\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}+\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {1}{7} x^7 S(b x)-\frac {6 x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {24 \operatorname {Subst}\left (\int \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^5 \pi ^3}\\ &=-\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}+\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {1}{7} x^7 S(b x)+\frac {48 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {6 x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.76 \[ -\frac {6 \left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac {x^2 \left (\pi ^2 b^4 x^4-24\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac {1}{7} x^7 S(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{6} {\rm fresnels}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 107, normalized size = 0.98 \[ \frac {\frac {b^{7} x^{7} \mathrm {S}\left (b x \right )}{7}+\frac {b^{6} x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }-\frac {6 \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }\right )}{7 \pi }}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.50, size = 156, normalized size = 1.43 \[ \frac {3 x^{7} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{28 \Gamma \left (\frac {7}{4}\right )} + \frac {3 x^{6} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{28 \pi b \Gamma \left (\frac {7}{4}\right )} - \frac {9 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{14 \pi ^{2} b^{3} \Gamma \left (\frac {7}{4}\right )} - \frac {18 x^{2} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{7 \pi ^{3} b^{5} \Gamma \left (\frac {7}{4}\right )} + \frac {36 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{7 \pi ^{4} b^{7} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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