Optimal. Leaf size=177 \[ \frac {43 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} \pi ^3 b^5}+\frac {2 x^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac {4 x^3}{15 \pi ^2 b^2}+\frac {x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}-\frac {16 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac {11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}-\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}+\frac {1}{5} x^5 S(b x)^2 \]
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Rubi [A] time = 0.19, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6430, 6454, 6462, 3391, 30, 3386, 3351, 6452, 3385} \[ -\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}+\frac {2 x^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}-\frac {16 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}+\frac {43 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} \pi ^3 b^5}+\frac {4 x^3}{15 \pi ^2 b^2}-\frac {11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}+\frac {x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}+\frac {1}{5} x^5 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 3351
Rule 3385
Rule 3386
Rule 3391
Rule 6430
Rule 6452
Rule 6454
Rule 6462
Rubi steps
\begin {align*} \int x^4 S(b x)^2 \, dx &=\frac {1}{5} x^5 S(b x)^2-\frac {1}{5} (2 b) \int x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b \pi }+\frac {1}{5} x^5 S(b x)^2-\frac {\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{5 \pi }-\frac {8 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{5 b \pi }\\ &=\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b \pi }+\frac {1}{5} x^5 S(b x)^2-\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}+\frac {16 \int x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b^3 \pi ^2}-\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{10 b^2 \pi ^2}+\frac {8 \int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b \pi }+\frac {1}{5} x^5 S(b x)^2-\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {3 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}+\frac {3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{20 b^4 \pi ^3}+\frac {8 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}+\frac {4 \int x^2 \, dx}{5 b^2 \pi ^2}-\frac {4 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=\frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b \pi }+\frac {1}{5} x^5 S(b x)^2+\frac {3 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {4 \sqrt {2} S\left (\sqrt {2} b x\right )}{5 b^5 \pi ^3}-\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}+\frac {2 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}\\ &=\frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{5 b \pi }+\frac {1}{5} x^5 S(b x)^2+\frac {3 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {\sqrt {2} S\left (\sqrt {2} b x\right )}{b^5 \pi ^3}-\frac {8 x^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 137, normalized size = 0.77 \[ \frac {24 \pi ^3 b^5 x^5 S(b x)^2+32 \pi b^3 x^3-66 b x \sin \left (\pi b^2 x^2\right )+48 S(b x) \left (\left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )-4 \pi b^2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )\right )+12 \pi b^3 x^3 \cos \left (\pi b^2 x^2\right )+129 \sqrt {2} S\left (\sqrt {2} b x\right )}{120 \pi ^3 b^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} {\rm fresnels}\left (b x\right )^{2}, x\right ) \]
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 \[ \int x^{4} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 208, normalized size = 1.18 \[ \frac {\frac {b^{5} x^{5} \mathrm {S}\left (b x \right )^{2}}{5}-2 \,\mathrm {S}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}-\frac {4 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{5 \pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{5 \pi ^{3}}}{b^{5}} \]
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 \[ \int x^{4} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
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 \[ \int x^4\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x^{4} S^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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