Optimal. Leaf size=104 \[ -\frac {x}{b^3 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}-\frac {5 \text {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi } \]
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Rubi [A]
time = 0.06, 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 = {6590, 6596,
3439, 3433, 3466} \begin {gather*} -\frac {5 \text {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}-\frac {x}{\pi ^2 b^3}+\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 {x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3439
Rule 3466
Rule 6590
Rule 6596
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 = 0.06, size = 83, normalized size = 0.80 \begin {gather*} \frac {2 b x \left (-4+\cos \left (b^2 \pi x^2\right )\right )-5 \sqrt {2} \text {FresnelC}\left (\sqrt {2} b x\right )+8 \text {FresnelC}(b x) \left (2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )}{8 b^4 \pi ^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.89, size = 114, normalized size = 1.10
method | result | size |
default | \(\frac {\frac {\FresnelC \left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {b x}{\pi ^{2}}+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi ^{2}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }}{2 \pi }}{b^{3}}}{b}\) | \(114\) |
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 = 94, normalized size = 0.90 \begin {gather*} \frac {8 \, \pi b^{3} x^{2} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 10 \, b^{2} x + 16 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{8 \, \pi ^{2} b^{5}} \end {gather*}
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 \begin {gather*} \int x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \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^3\,\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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