Optimal. Leaf size=231 \[ \frac {1}{48} \pi ^3 b^6 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac {1}{45} \sqrt {2} \pi ^3 b^6 C\left (\sqrt {2} b x\right )-\frac {7 \pi ^3 b^6 C\left (\sqrt {2} b x\right )}{144 \sqrt {2}}+\frac {\pi b^3}{144 x^3}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{24 x^4}-\frac {b \sin \left (\pi b^2 x^2\right )}{60 x^5}+\frac {67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{1440 x}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{48 x^2}-\frac {13 \pi b^3 \cos \left (\pi b^2 x^2\right )}{720 x^3} \]
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Rubi [A] time = 0.20, 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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{6} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {1}{48} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{24} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}-\frac {1}{96} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{72} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{48} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{36} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{45} \left (2 b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {7 b^6 \pi ^3 C\left (\sqrt {2} b x\right )}{144 \sqrt {2}}-\frac {1}{45} \sqrt {2} b^6 \pi ^3 C\left (\sqrt {2} b x\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}, x\right ) \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{7}}\, dx \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \]
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 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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