Optimal. Leaf size=74 \[ \frac {3 C(b x)}{4 \pi ^2 b^4}-\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 C(b x) \]
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Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6427, 3386, 3385, 3352} \[ \frac {3 \text {FresnelC}(b x)}{4 \pi ^2 b^4}-\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 \text {FresnelC}(b x) \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3385
Rule 3386
Rule 6427
Rubi steps
\begin {align*} \int x^3 C(b x) \, dx &=\frac {1}{4} x^4 C(b x)-\frac {1}{4} b \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{4} x^4 C(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b \pi }\\ &=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {1}{4} x^4 C(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 C(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 C(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }\\ \end {align*}
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Mathematica [A] time = 0.02, size = 74, normalized size = 1.00 \[ \frac {3 C(b x)}{4 \pi ^2 b^4}-\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 C(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} {\rm fresnelc}\left (b x\right ), x\right ) \]
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 \[ \int x^{3} {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.95 \[ \frac {\frac {b^{4} x^{4} \FresnelC \left (b x \right )}{4}-\frac {b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {3 \FresnelC \left (b x \right )}{4 \pi }}{\pi }}{b^{4}} \]
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 \[ \int x^{3} {\rm fresnelc}\left (b x\right )\,{d x} \]
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 \[ \int x^3\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.01, size = 112, normalized size = 1.51 \[ \frac {5 x^{4} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \Gamma \left (\frac {9}{4}\right )} - \frac {5 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi b \Gamma \left (\frac {9}{4}\right )} - \frac {15 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{3} \Gamma \left (\frac {9}{4}\right )} + \frac {15 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{4} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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