Optimal. Leaf size=224 \[ -\frac{\pi ^3 b^6 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}+\frac{\pi b^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{1}{210} \pi ^4 b^7 \text{FresnelC}(b x)^2+\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}+\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}+\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}-\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
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Rubi [A] time = 0.351595, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6457, 6465, 6441, 30, 3375, 3380, 3297, 3299, 3379} \[ -\frac{\pi ^3 b^6 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}+\frac{\pi b^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{1}{210} \pi ^4 b^7 \text{FresnelC}(b x)^2+\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}+\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}+\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}-\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
Antiderivative was successfully verified.
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Rule 6457
Rule 6465
Rule 6441
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rule 3379
Rubi steps
\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^8} \, dx &=-\frac{b}{84 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{1}{14} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{7} \left (b^2 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b}{84 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}+\frac{1}{28} b \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac{1}{70} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{35} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^4} \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}-\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{105 x^3}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{1}{140} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{84} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{210} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}-\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{105 x^3}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b^6 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x}+\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac{1}{420} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{280} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{168} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{210} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac{1}{105} \left (b^8 \pi ^4\right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}-\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{105 x^3}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b^6 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x}+\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac{1}{420} b^7 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{420} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{280} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{168} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{105} \left (b^7 \pi ^4\right ) \operatorname{Subst}(\int x \, dx,x,C(b x))\\ &=-\frac{b}{84 x^6}+\frac{b^5 \pi ^2}{420 x^2}-\frac{b \cos \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{105 x^3}+\frac{1}{210} b^7 \pi ^4 C(b x)^2+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b^6 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x}+\frac{b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac{1}{70} b^7 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0135264, size = 224, normalized size = 1. \[ -\frac{\pi ^3 b^6 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x}+\frac{\pi b^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{1}{210} \pi ^4 b^7 \text{FresnelC}(b x)^2+\frac{1}{70} \pi ^3 b^7 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^5}{420 x^2}+\frac{\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}+\frac{\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}-\frac{b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac{b}{84 x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{8}}\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 [F] 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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] 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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] 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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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