Optimal. Leaf size=166 \[ -\frac {43 C\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}+\frac {4 x}{\pi ^3 b^5}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {11 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {4 x^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^3 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^5}{10 \pi b} \]
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Rubi [A] time = 0.17, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6462, 3391, 30, 3386, 3385, 3352, 6454, 6460, 3357} \[ -\frac {43 \text {FresnelC}\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {4 x^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^3 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {11 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {4 x}{\pi ^3 b^5}-\frac {x^5}{10 \pi b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3352
Rule 3357
Rule 3385
Rule 3386
Rule 3391
Rule 6454
Rule 6460
Rule 6462
Rubi steps
\begin {align*} \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {4 \int x^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^4 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {8 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^4 \pi ^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\int x^4 \, dx}{2 b \pi }+\frac {\int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^5}{10 b \pi }+\frac {x \cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}+\frac {8 \int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {3 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {C\left (\sqrt {2} b x\right )}{\sqrt {2} b^6 \pi ^3}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {3 \int \cos \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac {8 \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^5 \pi ^3}\\ &=\frac {4 x}{b^5 \pi ^3}-\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {11 C\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {4 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=\frac {4 x}{b^5 \pi ^3}-\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {11 C\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}-\frac {2 \sqrt {2} C\left (\sqrt {2} b x\right )}{b^6 \pi ^3}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac {8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 126, normalized size = 0.76 \[ \frac {80 S(b x) \left (4 \pi b^2 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )+\left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )\right )+2 b x \left (-4 \pi ^2 b^4 x^4+10 \pi b^2 x^2 \sin \left (\pi b^2 x^2\right )+55 \cos \left (\pi b^2 x^2\right )+160\right )-215 \sqrt {2} C\left (\sqrt {2} b x\right )}{80 \pi ^3 b^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{5} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right ), x\right ) \]
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 \[ \int x^{5} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 212, normalized size = 1.28 \[ \frac {\frac {\mathrm {S}\left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }\right )}{b^{5}}-\frac {\frac {\frac {1}{5} \pi ^{2} b^{5} x^{5}-8 b x}{2 \pi ^{3}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi }}{\pi ^{2}}-\frac {\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi ^{3}}}{b^{5}}}{b} \]
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 \[ \int x^{5} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )\,{d x} \]
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 \[ \int x^5\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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