Optimal. Leaf size=224 \[ \frac {1}{210} \pi ^4 b^7 S(b x)^2+\frac {\pi ^2 b^5}{420 x^2}-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac {b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac {1}{70} \pi ^3 b^7 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^3 b^6 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x}-\frac {\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac {\pi ^2 b^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac {b}{84 x^6} \]
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Rubi [A] time = 0.35, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6456, 6464, 6440, 30, 3375, 3380, 3297, 3299, 3379} \[ \frac {\pi ^2 b^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac {\pi ^3 b^6 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x}-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac {1}{210} \pi ^4 b^7 S(b x)^2-\frac {1}{70} \pi ^3 b^7 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^2 b^5}{420 x^2}-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac {\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac {b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac {b}{84 x^6} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3297
Rule 3299
Rule 3375
Rule 3379
Rule 3380
Rule 6440
Rule 6456
Rule 6464
Rubi steps
\begin {align*} \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx &=-\frac {b}{84 x^6}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{14} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac {1}{7} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^6} \, dx\\ &=-\frac {b}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{28} b \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{70} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{35} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}+\frac {1}{140} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{84} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{210} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx\\ &=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac {1}{420} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{280} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{168} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{105} \left (b^8 \pi ^4\right ) \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{420} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{420} \left (b^7 \pi ^3\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{280} \left (b^7 \pi ^3\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{168} \left (b^7 \pi ^3\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{105} \left (b^7 \pi ^4\right ) \operatorname {Subst}(\int x \, dx,x,S(b x))\\ &=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 S(b x)^2-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 224, normalized size = 1.00 \[ \frac {1}{210} \pi ^4 b^7 S(b x)^2+\frac {\pi ^2 b^5}{420 x^2}-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac {b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac {1}{70} \pi ^3 b^7 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^3 b^6 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x}-\frac {\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac {\pi ^2 b^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac {b}{84 x^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \]
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 \[ \int \frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{8}}\, dx \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \]
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 \[ \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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