Integrand size = 10, antiderivative size = 265 \[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\frac {5 x^4}{24 b^2 \pi ^2}+\frac {11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}-\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3} \]
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Time = 0.21 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6566, 6590, 6598, 6582, 3460, 2718, 3461, 3390, 30, 3377} \[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi ^3 b^6}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {5 x^4}{24 \pi ^2 b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}+\frac {11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}-\frac {5 x^3 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2 \]
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Rule 30
Rule 2718
Rule 3377
Rule 3390
Rule 3460
Rule 3461
Rule 6566
Rule 6582
Rule 6590
Rule 6598
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {1}{3} b \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx \\ & = \frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {\int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{6 \pi }+\frac {5 \int x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b \pi } \\ & = -\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {5 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^3 \pi ^2}+\frac {5 \int x^3 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}+\frac {\text {Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 \pi } \\ & = -\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }-\frac {5 \int \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {5 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^4 \pi ^3}+\frac {\text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2}+\frac {5 \text {Subst}\left (\int x \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2} \\ & = -\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {x^2 \sin \left (b^2 \pi x^2\right )}{6 b^4 \pi ^3}-\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^4 \pi ^3}-\frac {5 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^4 \pi ^3}+\frac {5 \text {Subst}\left (\int x \, dx,x,x^2\right )}{12 b^2 \pi ^2}+\frac {5 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^2 \pi ^2} \\ & = \frac {5 x^4}{24 b^2 \pi ^2}+\frac {17 \cos \left (b^2 \pi x^2\right )}{12 b^6 \pi ^4}-\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3}-\frac {5 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^4 \pi ^3} \\ & = \frac {5 x^4}{24 b^2 \pi ^2}+\frac {11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}-\frac {x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)^2-\frac {5 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}-\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {5 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3} \\ \end{align*}
\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^5 \operatorname {FresnelC}(b x)^2 \, dx \]
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\[\int x^{5} \operatorname {FresnelC}\left (b x \right )^{2}d x\]
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\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]
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\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{5} C^{2}\left (b x\right )\, dx \]
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\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]
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\[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{5} \operatorname {C}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x^5 \operatorname {FresnelC}(b x)^2 \, dx=\int x^5\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]
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