Optimal. Leaf size=232 \[ \frac {105 S(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}-\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rubi [A] time = 0.38, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6454, 6462, 3379, 3309, 30, 3296, 2637, 2634, 6440} \[ \frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {105 S(b x)^2}{2 \pi ^4 b^9}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {105 x^2}{4 \pi ^4 b^7}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2634
Rule 2637
Rule 3296
Rule 3309
Rule 3379
Rule 6440
Rule 6454
Rule 6462
Rubi steps
\begin {align*} \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)}{b^2 \pi }+\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^2 \pi }+\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {35 \int x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac {7 \int x^5 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac {\operatorname {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {105 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^6 \pi ^3}-\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}+\frac {3 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \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 )}{2 b^3 \pi ^2}\\ &=-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {105 \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^8 \pi ^4}+\frac {105 \int x \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}-\frac {3 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac {35 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {7 \operatorname {Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b^3 \pi ^2}+\frac {7 \operatorname {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}\\ &=-\frac {7 x^6}{12 b^3 \pi ^2}+\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {105 \operatorname {Subst}(\int x \, dx,x,S(b x))}{b^9 \pi ^4}-\frac {3 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}-\frac {35 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}+\frac {105 \operatorname {Subst}\left (\int \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}-\frac {7 \operatorname {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}\\ &=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {105 S(b x)^2}{2 b^9 \pi ^4}-\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {73 \sin \left (b^2 \pi x^2\right )}{2 b^9 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {7 \operatorname {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}\\ &=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}+\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {105 S(b x)^2}{2 b^9 \pi ^4}-\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 232, normalized size = 1.00 \[ \frac {105 S(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}-\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}+\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{8} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]
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 \[ \int x^{8} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{8} \mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )\, dx \]
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 \[ \int x^{8} {\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^8\,\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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