Optimal. Leaf size=278 \[ -\frac{\pi ^4 b^8 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac{\pi ^3 b^6 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}-\frac{\pi b^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}+\frac{\pi ^5 b^9 \text{FresnelC}(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}+\frac{\pi ^3 b^7}{3780 x^2}-\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
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Rubi [A] time = 0.520338, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6465, 6457, 6441, 30, 3375, 3380, 3297, 3299, 3379} \[ -\frac{\pi ^4 b^8 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac{\pi ^3 b^6 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}-\frac{\pi b^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}+\frac{\pi ^5 b^9 \text{FresnelC}(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}+\frac{\pi ^3 b^7}{3780 x^2}-\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
Antiderivative was successfully verified.
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Rule 6465
Rule 6457
Rule 6441
Rule 30
Rule 3375
Rule 3380
Rule 3297
Rule 3299
Rule 3379
Rubi steps
\begin{align*} \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{1}{18} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^9} \, dx+\frac{1}{9} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^8} \, dx\\ &=-\frac{b^3 \pi }{756 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{1}{36} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )+\frac{1}{126} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{63} \left (b^4 \pi ^2\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b^3 \pi }{756 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{1}{252} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{144} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac{1}{630} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{315} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^4} \, dx\\ &=-\frac{b^3 \pi }{756 x^6}+\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{945 x^3}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}-\frac{\left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )}{1260}-\frac{1}{756} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{432} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{\left (b^7 \pi ^3\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac{1}{945} \left (b^8 \pi ^4\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b^3 \pi }{756 x^6}+\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{945 x^3}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{b^8 \pi ^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3780}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{2520}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1512}-\frac{1}{864} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{\left (b^9 \pi ^4\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx}{1890}+\frac{1}{945} \left (b^{10} \pi ^5\right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac{b^3 \pi }{756 x^6}+\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{945 x^3}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{b^8 \pi ^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac{b^9 \pi ^4 \text{Si}\left (b^2 \pi x^2\right )}{3780}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{3780}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{2520}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1512}+\frac{1}{864} \left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac{1}{945} \left (b^9 \pi ^5\right ) \operatorname{Subst}(\int x \, dx,x,C(b x))\\ &=-\frac{b^3 \pi }{756 x^6}+\frac{b^7 \pi ^3}{3780 x^2}-\frac{11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{945 x^3}+\frac{b^9 \pi ^5 C(b x)^2}{1890}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{b^8 \pi ^4 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{945 x}-\frac{b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac{67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac{83 b^9 \pi ^4 \text{Si}\left (b^2 \pi x^2\right )}{30240}\\ \end{align*}
Mathematica [A] time = 0.0174078, size = 278, normalized size = 1. \[ -\frac{\pi ^4 b^8 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x}+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac{\pi ^3 b^6 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}-\frac{\pi b^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}+\frac{\pi ^5 b^9 \text{FresnelC}(b x)^2}{1890}+\frac{83 \pi ^4 b^9 \text{Si}\left (b^2 \pi x^2\right )}{30240}+\frac{\pi ^3 b^7}{3780 x^2}-\frac{\pi b^3}{756 x^6}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{30240 x^4}-\frac{b \sin \left (\pi b^2 x^2\right )}{144 x^8}+\frac{5 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3024 x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{10}}\sin \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{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{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 ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}, 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{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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