Optimal. Leaf size=104 \[ -\frac {5 C\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}-\frac {x}{\pi ^2 b^3}+\frac {x^2 C(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 C(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6455, 6461, 3358, 3352, 3385} \[ \frac {x^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {5 \text {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}+\frac {x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x}{\pi ^2 b^3} \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3358
Rule 3385
Rule 6455
Rule 6461
Rubi steps
\begin {align*} \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac {x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {2 \int x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {2 \int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {C\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {2 \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {x}{b^3 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {C\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {x}{b^3 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {5 C\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }\\ \end {align*}
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Mathematica [A] time = 0.10, size = 83, normalized size = 0.80 \[ \frac {8 C(b x) \left (\pi b^2 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )+2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )\right )+2 b x \left (\cos \left (\pi b^2 x^2\right )-4\right )-5 \sqrt {2} C\left (\sqrt {2} b x\right )}{8 \pi ^2 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right ), x\right ) \]
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 \[ \int x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 114, normalized size = 1.10 \[ \frac {\frac {\FresnelC \left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {b x}{\pi ^{2}}+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi ^{2}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }}{2 \pi }}{b^{3}}}{b} \]
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 \[ \int x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnelc}\left (b x\right )\,{d x} \]
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 \[ \int x^3\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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 \[ \int x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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