Optimal. Leaf size=158 \[ -\frac {43 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}-\frac {2 x^3}{3 \pi ^2 b^3}-\frac {x^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {8 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {11 x \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {4 x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rubi [A] time = 0.16, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6454, 6462, 3391, 30, 3386, 3351, 6452, 3385} \[ \frac {4 x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {8 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {43 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}-\frac {2 x^3}{3 \pi ^2 b^3}+\frac {11 x \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3351
Rule 3385
Rule 3386
Rule 3391
Rule 6452
Rule 6454
Rule 6462
Rubi steps
\begin {align*} \int x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^2 \pi }+\frac {\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {4 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {8 \int x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}+\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {4 \int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {4 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {3 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}-\frac {4 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {2 \int x^2 \, dx}{b^3 \pi ^2}+\frac {2 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {2 x^3}{3 b^3 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {3 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}-\frac {2 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^6 \pi ^3}+\frac {4 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {11 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=-\frac {2 x^3}{3 b^3 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {11 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}-\frac {2 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^6 \pi ^3}+\frac {4 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {11 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 120, normalized size = 0.76 \[ -\frac {32 \pi b^3 x^3-66 b x \sin \left (\pi b^2 x^2\right )+48 S(b x) \left (\left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )-4 \pi b^2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )\right )+12 \pi b^3 x^3 \cos \left (\pi b^2 x^2\right )+129 \sqrt {2} S\left (\sqrt {2} b x\right )}{48 \pi ^3 b^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{5} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]
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 \[ \int x^{5} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 202, normalized size = 1.28 \[ \frac {\frac {\mathrm {S}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{b^{5}}-\frac {\frac {2 b^{3} x^{3}}{3 \pi ^{2}}-\frac {2 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{\pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi ^{3}}}{b^{5}}}{b} \]
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 \[ \int x^{5} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
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 \[ \int x^5\,\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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 \[ \int x^{5} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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