Optimal. Leaf size=152 \[ -\frac{1}{8} \pi ^2 b^4 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac{\pi b^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 x^2}-\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac{7 \pi ^2 b^4 \text{FresnelC}\left (\sqrt{2} b x\right )}{24 \sqrt{2}}+\frac{7 \pi b^3 \sin \left (\pi b^2 x^2\right )}{48 x}-\frac{b \cos \left (\pi b^2 x^2\right )}{24 x^3}-\frac{b}{24 x^3} \]
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Rubi [A] time = 0.115349, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.0300349, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) \text{FresnelC}(b x)}{x^5} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{5}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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