Optimal. Leaf size=153 \[ -\frac {1}{8} \pi ^2 b^4 \text {Int}\left (\frac {C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac {7 \pi ^2 b^4 C\left (\sqrt {2} b x\right )}{24 \sqrt {2}}+\frac {\pi b^2 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^2}-\frac {C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b \cos \left (\pi b^2 x^2\right )}{24 x^3}+\frac {7 \pi b^3 \sin \left (\pi b^2 x^2\right )}{48 x}-\frac {b}{24 x^3} \]
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Rubi [A] time = 0.12, 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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^5} \, dx \]
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 ) C(b x)}{x^5} \, dx &=-\frac {b}{24 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac {1}{8} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{4} \left (b^2 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac {b}{24 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^2}-\frac {1}{16} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ &=-\frac {b}{24 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^2}+\frac {7 b^3 \pi \sin \left (b^2 \pi x^2\right )}{48 x}-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx-\frac {1}{8} \left (b^5 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{6} \left (b^5 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b}{24 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}-\frac {7 b^4 \pi ^2 C\left (\sqrt {2} b x\right )}{24 \sqrt {2}}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^2}+\frac {7 b^3 \pi \sin \left (b^2 \pi x^2\right )}{48 x}-\frac {1}{8} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^5} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \]
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 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{x^{5}}\, dx \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^5} \,d x \]
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 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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