Optimal. Leaf size=120 \[ -\frac {3 x^2}{4 b^3 \pi ^2}+\frac {x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}-\frac {3 \text {FresnelC}(b x)^2}{2 b^5 \pi ^2}+\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\sin \left (b^2 \pi x^2\right )}{b^5 \pi ^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6590, 6598,
6576, 30, 3461, 2714, 3460, 3377, 2717} \begin {gather*} -\frac {3 \text {FresnelC}(b x)^2}{2 \pi ^2 b^5}-\frac {3 x^2}{4 \pi ^2 b^3}+\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\sin \left (\pi b^2 x^2\right )}{\pi ^3 b^5}+\frac {3 x \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2714
Rule 2717
Rule 3377
Rule 3460
Rule 3461
Rule 6576
Rule 6590
Rule 6598
Rubi steps
\begin {align*} \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \int x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^4 \pi ^2}-\frac {3 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=\frac {x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 \text {Subst}(\int x \, dx,x,C(b x))}{b^5 \pi ^2}-\frac {\text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {3 \text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac {3 x^2}{4 b^3 \pi ^2}+\frac {x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {3 C(b x)^2}{2 b^5 \pi ^2}+\frac {x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\sin \left (b^2 \pi x^2\right )}{b^5 \pi ^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 120, normalized size = 1.00 \begin {gather*} -\frac {3 x^2}{4 b^3 \pi ^2}+\frac {x^2 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}-\frac {3 \text {FresnelC}(b x)^2}{2 b^5 \pi ^2}+\frac {x^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\sin \left (b^2 \pi x^2\right )}{b^5 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 105, normalized size = 0.88 \begin {gather*} \frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, \pi b^{2} x^{2} + 6 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 3 \, \pi \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{2 \, \pi ^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 151, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac {x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{2} b^{3}} - \frac {x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} + \frac {3 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {2 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac {3 C^{2}\left (b x\right )}{2 \pi ^{2} b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^4\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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