Optimal. Leaf size=241 \[ -\frac {b}{60 x^5}+\frac {b^5 \pi ^2}{96 x}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{1440 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{48 x^2}+\frac {7 b^6 \pi ^3 S\left (\sqrt {2} b x\right )}{144 \sqrt {2}}+\frac {1}{45} \sqrt {2} b^6 \pi ^3 S\left (\sqrt {2} b x\right )+\frac {b^2 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}+\frac {13 b^3 \pi \sin \left (b^2 \pi x^2\right )}{720 x^3}+\frac {1}{48} b^6 \pi ^3 \text {Int}\left (\frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]
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Rubi [A]
time = 0.14, 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 = {}
\begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^7} \, dx \end {gather*}
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 ) C(b x)}{x^7} \, dx &=-\frac {b}{60 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{6 x^6}+\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{6} \left (b^2 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac {b}{60 x^5}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{6 x^6}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {1}{48} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\sin \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 ) C(b x)}{x^3} \, dx\\ &=-\frac {b}{60 x^5}+\frac {b^5 \pi ^2}{96 x}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{48 x^2}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}+\frac {13 b^3 \pi \sin \left (b^2 \pi x^2\right )}{720 x^3}-\frac {1}{96} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{72} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac {b}{60 x^5}+\frac {b^5 \pi ^2}{96 x}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{1440 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{48 x^2}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}+\frac {13 b^3 \pi \sin \left (b^2 \pi x^2\right )}{720 x^3}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {C(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 \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{36} \left (b^7 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{45} \left (2 b^7 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b}{60 x^5}+\frac {b^5 \pi ^2}{96 x}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{1440 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{48 x^2}+\frac {7 b^6 \pi ^3 S\left (\sqrt {2} b x\right )}{144 \sqrt {2}}+\frac {1}{45} \sqrt {2} b^6 \pi ^3 S\left (\sqrt {2} b x\right )+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}+\frac {13 b^3 \pi \sin \left (b^2 \pi x^2\right )}{720 x^3}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^7} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{x^{7}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{7}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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