Optimal. Leaf size=253 \[ -\frac {105 S(b x)^2}{8 \pi ^4 b^8}-\frac {105 x^2}{16 \pi ^4 b^6}+\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {7 x^6}{48 \pi ^2 b^2}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}+\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}-\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{8} x^8 S(b x)^2 \]
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Rubi [A] time = 0.43, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6430, 6454, 6462, 3379, 3309, 30, 3296, 2637, 2634, 6440} \[ -\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}-\frac {105 S(b x)^2}{8 \pi ^4 b^8}+\frac {7 x^6}{48 \pi ^2 b^2}-\frac {105 x^2}{16 \pi ^4 b^6}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}+\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}+\frac {1}{8} x^8 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2634
Rule 2637
Rule 3296
Rule 3309
Rule 3379
Rule 6430
Rule 6440
Rule 6454
Rule 6462
Rubi steps
\begin {align*} \int x^7 S(b x)^2 \, dx &=\frac {1}{8} x^8 S(b x)^2-\frac {1}{4} b \int x^8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{8 \pi }-\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{4 b \pi }\\ &=\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 \int x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}+\frac {7 \int x^5 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^2 \pi ^2}-\frac {\operatorname {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 \pi }\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {105 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{4 b^5 \pi ^3}+\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^4 \pi ^3}-\frac {3 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}+\frac {7 \operatorname {Subst}\left (\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{16 b^4 \pi ^3}-\frac {105 \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}-\frac {105 \int x \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^6 \pi ^4}+\frac {3 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}+\frac {35 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^4 \pi ^3}+\frac {7 \operatorname {Subst}\left (\int x^2 \, dx,x,x^2\right )}{16 b^2 \pi ^2}-\frac {7 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}\\ &=\frac {7 x^6}{48 b^2 \pi ^2}-\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {105 \operatorname {Subst}(\int x \, dx,x,S(b x))}{4 b^8 \pi ^4}+\frac {3 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}+\frac {35 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^6 \pi ^4}-\frac {105 \operatorname {Subst}\left (\int \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}+\frac {7 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {73 \sin \left (b^2 \pi x^2\right )}{8 b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {7 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 253, normalized size = 1.00 \[ -\frac {105 S(b x)^2}{8 \pi ^4 b^8}-\frac {105 x^2}{16 \pi ^4 b^6}+\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {7 x^6}{48 \pi ^2 b^2}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}+\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}-\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{8} x^8 S(b x)^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{7} {\rm fresnels}\left (b x\right )^{2}, x\right ) \]
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 \[ \int x^{7} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{7} \mathrm {S}\left (b x \right )^{2}\, dx \]
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 \[ \int x^{7} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^7\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x^{7} S^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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