Optimal. Leaf size=124 \[ \frac {5 C\left (\sqrt {2} b x\right )}{6 \sqrt {2} \pi ^2 b^3}-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {x \cos \left (\pi b^2 x^2\right )}{6 \pi ^2 b^2}+\frac {2 x}{3 \pi ^2 b^2}-\frac {4 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 C(b x)^2 \]
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Rubi [A] time = 0.11, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6431, 6455, 6461, 3358, 3352, 3385} \[ -\frac {2 x^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac {4 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {5 \text {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} \pi ^2 b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{6 \pi ^2 b^2}+\frac {2 x}{3 \pi ^2 b^2}+\frac {1}{3} x^3 \text {FresnelC}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3358
Rule 3385
Rule 6431
Rule 6455
Rule 6461
Rubi steps
\begin {align*} \int x^2 C(b x)^2 \, dx &=\frac {1}{3} x^3 C(b x)^2-\frac {1}{3} (2 b) \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{3} x^3 C(b x)^2-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{3 \pi }+\frac {4 \int x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b \pi }\\ &=-\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{3} x^3 C(b x)^2-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{6 b^2 \pi ^2}+\frac {4 \int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}\\ &=-\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{3} x^3 C(b x)^2+\frac {C\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {4 \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{3 b^2 \pi ^2}\\ &=\frac {2 x}{3 b^2 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{3} x^3 C(b x)^2+\frac {C\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }+\frac {2 \int \cos \left (b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}\\ &=\frac {2 x}{3 b^2 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{3 b^3 \pi ^2}+\frac {1}{3} x^3 C(b x)^2+\frac {C\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}+\frac {\sqrt {2} C\left (\sqrt {2} b x\right )}{3 b^3 \pi ^2}-\frac {2 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi }\\ \end {align*}
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Mathematica [A] time = 0.09, size = 100, normalized size = 0.81 \[ \frac {4 \pi ^2 b^3 x^3 C(b x)^2-8 C(b x) \left (\pi b^2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )+2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )\right )-2 b x \left (\cos \left (\pi b^2 x^2\right )-4\right )+5 \sqrt {2} C\left (\sqrt {2} b x\right )}{12 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} {\rm fresnelc}\left (b x\right )^{2}, x\right ) \]
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 \[ \int x^{2} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 0.98 \[ \frac {\frac {b^{3} x^{3} \FresnelC \left (b x \right )^{2}}{3}-2 \FresnelC \left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}\right )+\frac {2 b x}{3 \pi ^{2}}+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{3 \pi ^{2}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }}{3 \pi }}{b^{3}} \]
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 \[ \int x^{2} {\rm fresnelc}\left (b x\right )^{2}\,{d x} \]
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 \[ \int x^2\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x^{2} C^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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