Optimal. Leaf size=84 \[ -\frac {x^4 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac {8 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac {4 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}+\frac {1}{5} x^5 C(b x) \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6427, 3380, 3296, 2637} \[ -\frac {x^4 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac {8 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac {4 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}+\frac {1}{5} x^5 \text {FresnelC}(b x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3380
Rule 6427
Rubi steps
\begin {align*} \int x^4 C(b x) \, dx &=\frac {1}{5} x^5 C(b x)-\frac {1}{5} b \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{5} x^5 C(b x)-\frac {1}{10} b \operatorname {Subst}\left (\int x^2 \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 C(b x)-\frac {x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {2 \operatorname {Subst}\left (\int x \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{5 b \pi }\\ &=-\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)-\frac {x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {4 \operatorname {Subst}\left (\int \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{5 b^3 \pi ^2}\\ &=-\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)+\frac {8 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }\\ \end {align*}
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Mathematica [A] time = 0.06, size = 71, normalized size = 0.85 \[ -\frac {4 x^2 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}-\frac {\left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}+\frac {1}{5} x^5 C(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} {\rm fresnelc}\left (b x\right ), x\right ) \]
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 \[ \int x^{4} {\rm fresnelc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 81, normalized size = 0.96 \[ \frac {\frac {b^{5} x^{5} \FresnelC \left (b x \right )}{5}-\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {-\frac {4 b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }}{b^{5}} \]
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 \[ \int x^{4} {\rm fresnelc}\left (b x\right )\,{d x} \]
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 \[ \int x^4\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 116, normalized size = 1.38 \[ \frac {x^{5} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{20 \Gamma \left (\frac {5}{4}\right )} - \frac {x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{20 \pi b \Gamma \left (\frac {5}{4}\right )} - \frac {x^{2} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{5 \pi ^{2} b^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {2 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{5 \pi ^{3} b^{5} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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