Optimal. Leaf size=144 \[ \frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }+\frac {C(b x) S(b x)}{2 \pi b^2}-\frac {x C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 C(b x)^2 \]
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Rubi [A] time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6431, 6455, 6447, 3379, 2638} \[ \frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }+\frac {\text {FresnelC}(b x) S(b x)}{2 \pi b^2}-\frac {x \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^2}+\frac {1}{2} x^2 \text {FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3379
Rule 6431
Rule 6447
Rule 6455
Rubi steps
\begin {align*} \int x C(b x)^2 \, dx &=\frac {1}{2} x^2 C(b x)^2-b \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{2} x^2 C(b x)^2-\frac {x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+\frac {\int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 \pi }+\frac {\int C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac {1}{2} x^2 C(b x)^2+\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+\frac {\operatorname {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 \pi }\\ &=-\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {1}{2} x^2 C(b x)^2+\frac {C(b x) S(b x)}{2 b^2 \pi }+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int x C(b x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x {\rm fresnelc}\left (b x\right )^{2}, x\right ) \]
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 \[ \int x {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
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 \[ \int x \FresnelC \left (b x \right )^{2}\, dx \]
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 \[ \int x {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x C^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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