Optimal. Leaf size=242 \[ -\frac{\pi ^3 b^7 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x}+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{140 x^5}+\frac{\pi ^2 b^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x^3}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{28 x^7}+\frac{1}{840} \pi ^4 b^8 \text{FresnelC}(b x)^2+\frac{1}{280} \pi ^3 b^8 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^6}{1680 x^2}-\frac{b^2}{336 x^6}+\frac{\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}+\frac{\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}-\frac{\text{FresnelC}(b x)^2}{8 x^8} \]
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Rubi [A] time = 0.388084, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6431, 6457, 6465, 6441, 30, 3375, 3380, 3297, 3299, 3379} \[ -\frac{\pi ^3 b^7 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x}+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{140 x^5}+\frac{\pi ^2 b^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x^3}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{28 x^7}+\frac{1}{840} \pi ^4 b^8 \text{FresnelC}(b x)^2+\frac{1}{280} \pi ^3 b^8 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^6}{1680 x^2}-\frac{b^2}{336 x^6}+\frac{\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}+\frac{\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}-\frac{\text{FresnelC}(b x)^2}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 6431
Rule 6457
Rule 6465
Rule 6441
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rule 3379
Rubi steps
\begin{align*} \int \frac{C(b x)^2}{x^9} \, dx &=-\frac{C(b x)^2}{8 x^8}+\frac{1}{4} b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^8} \, dx\\ &=-\frac{b^2}{336 x^6}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac{C(b x)^2}{8 x^8}+\frac{1}{56} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{28} \left (b^3 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b^2}{336 x^6}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}-\frac{C(b x)^2}{8 x^8}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{140 x^5}+\frac{1}{112} b^2 \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac{1}{280} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{140} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^4} \, dx\\ &=-\frac{b^2}{336 x^6}+\frac{b^6 \pi ^2}{1680 x^2}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac{C(b x)^2}{8 x^8}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac{1}{560} \left (b^4 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{336} \left (b^4 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{840} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac{1}{420} \left (b^7 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b^2}{336 x^6}+\frac{b^6 \pi ^2}{1680 x^2}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac{C(b x)^2}{8 x^8}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{420 x}+\frac{b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac{\left (b^6 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1680}-\frac{\left (b^6 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1120}-\frac{1}{672} \left (b^6 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{840} \left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac{1}{420} \left (b^9 \pi ^4\right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac{b^2}{336 x^6}+\frac{b^6 \pi ^2}{1680 x^2}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac{b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}-\frac{C(b x)^2}{8 x^8}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{420 x}+\frac{b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac{b^8 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )}{1680}+\frac{\left (b^8 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1680}+\frac{\left (b^8 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1120}+\frac{1}{672} \left (b^8 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{420} \left (b^8 \pi ^4\right ) \operatorname{Subst}(\int x \, dx,x,C(b x))\\ &=-\frac{b^2}{336 x^6}+\frac{b^6 \pi ^2}{1680 x^2}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}+\frac{b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{28 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{420 x^3}+\frac{1}{840} b^8 \pi ^4 C(b x)^2-\frac{C(b x)^2}{8 x^8}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{140 x^5}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{420 x}+\frac{b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac{1}{280} b^8 \pi ^3 \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0126167, size = 242, normalized size = 1. \[ -\frac{\pi ^3 b^7 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x}+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{140 x^5}+\frac{\pi ^2 b^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{420 x^3}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{28 x^7}+\frac{1}{840} \pi ^4 b^8 \text{FresnelC}(b x)^2+\frac{1}{280} \pi ^3 b^8 \text{Si}\left (b^2 \pi x^2\right )+\frac{\pi ^2 b^6}{1680 x^2}-\frac{b^2}{336 x^6}+\frac{\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}+\frac{\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}-\frac{\text{FresnelC}(b x)^2}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}}{{x}^{9}}}\, 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{{\rm fresnelc}\left (b x\right )^{2}}{x^{9}}\,{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{{\rm fresnelc}\left (b x\right )^{2}}{x^{9}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C^{2}\left (b x\right )}{x^{9}}\, dx \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{{\rm fresnelc}\left (b x\right )^{2}}{x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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