Optimal. Leaf size=148 \[ -\frac {1}{15} \pi ^2 b^4 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x^3}+\frac {b \cos \left (\pi b^2 x^2\right )}{40 x^4}+\frac {1}{24} \pi ^2 b^5 \text {Ci}\left (b^2 \pi x^2\right )-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{24 x^2}-\frac {b}{40 x^4} \]
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Rubi [A] time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx &=-\frac {b}{40 x^4}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac {1}{10} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac {1}{5} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx\\ &=-\frac {b}{40 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac {1}{20} b \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b}{40 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 x^5}+\frac {1}{60} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{40} \left (b^3 \pi \right ) \operatorname {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b}{40 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{60} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{40} \left (b^5 \pi ^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b}{40 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}+\frac {1}{24} b^5 \pi ^2 \text {Ci}\left (b^2 \pi x^2\right )-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{15 x^3}-\frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 x^5}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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