Optimal. Leaf size=265 \[ -\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}-\frac {5 C(b x) S(b x)}{2 \pi ^3 b^6}-\frac {x^5 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {5 x^4}{24 \pi ^2 b^2}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}+\frac {11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}+\frac {5 x C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}-\frac {5 x^3 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{6} x^6 C(b x)^2 \]
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Rubi [A] time = 0.29, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6431, 6455, 6463, 6447, 3379, 2638, 3380, 3309, 30, 3296} \[ -\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac {5 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}-\frac {5 \text {FresnelC}(b x) S(b x)}{2 \pi ^3 b^6}-\frac {x^5 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {5 x \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}-\frac {5 x^3 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {5 x^4}{24 \pi ^2 b^2}+\frac {7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}-\frac {x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}+\frac {11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}+\frac {1}{6} x^6 \text {FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2638
Rule 3296
Rule 3309
Rule 3379
Rule 3380
Rule 6431
Rule 6447
Rule 6455
Rule 6463
Rubi steps
\begin {align*} \int x^5 C(b x)^2 \, dx &=\frac {1}{6} x^6 C(b x)^2-\frac {1}{3} b \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{6} x^6 C(b x)^2-\frac {x^5 C(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 C(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 ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 C(b x)^2-\frac {x^5 C(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 ) C(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 {\operatorname {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 ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 C(b x)^2+\frac {5 x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }-\frac {5 \int C(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 {\operatorname {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2}+\frac {5 \operatorname {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 ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 C(b x)^2-\frac {5 C(b x) S(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 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 C(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 {\operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^4 \pi ^3}-\frac {5 \operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^4 \pi ^3}+\frac {5 \operatorname {Subst}\left (\int x \, dx,x,x^2\right )}{12 b^2 \pi ^2}+\frac {5 \operatorname {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 ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 C(b x)^2-\frac {5 C(b x) S(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 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 C(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 \operatorname {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 ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{6} x^6 C(b x)^2-\frac {5 C(b x) S(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 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^5 \pi ^3}-\frac {x^5 C(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*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int x^5 C(b x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{5} {\rm fresnelc}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int x^{5} \FresnelC \left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^5\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} C^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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