Optimal. Leaf size=177 \[ \frac {4 x^3}{15 b^2 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 \text {FresnelC}(b x)^2-\frac {43 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {16 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {2 x^4 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6566, 6590,
6598, 6588, 3432, 3473, 30, 3467, 3466} \begin {gather*} -\frac {43 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} \pi ^3 b^5}-\frac {2 x^4 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac {4 x^3}{15 \pi ^2 b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}+\frac {16 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}+\frac {11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}-\frac {8 x^2 \text {FresnelC}(b x) \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)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3432
Rule 3466
Rule 3467
Rule 3473
Rule 6566
Rule 6588
Rule 6590
Rule 6598
Rubi steps
\begin {align*} \int x^4 C(b x)^2 \, dx &=\frac {1}{5} x^5 C(b x)^2-\frac {1}{5} (2 b) \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac {1}{5} x^5 C(b x)^2-\frac {2 x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{5 \pi }+\frac {8 \int x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b \pi }\\ &=-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)^2-\frac {2 x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {16 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{5 b^3 \pi ^2}+\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{10 b^2 \pi ^2}+\frac {8 \int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)^2+\frac {16 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {2 x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {3 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}-\frac {3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{20 b^4 \pi ^3}-\frac {8 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}+\frac {4 \int x^2 \, dx}{5 b^2 \pi ^2}+\frac {4 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2}\\ &=\frac {4 x^3}{15 b^2 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)^2-\frac {3 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}-\frac {4 \sqrt {2} S\left (\sqrt {2} b x\right )}{5 b^5 \pi ^3}+\frac {16 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {2 x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}-\frac {2 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}\\ &=\frac {4 x^3}{15 b^2 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 C(b x)^2-\frac {3 S\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}-\frac {\sqrt {2} S\left (\sqrt {2} b x\right )}{b^5 \pi ^3}+\frac {16 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {2 x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 137, normalized size = 0.77 \begin {gather*} \frac {32 b^3 \pi x^3-12 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+24 b^5 \pi ^3 x^5 \text {FresnelC}(b x)^2-129 \sqrt {2} S\left (\sqrt {2} b x\right )-48 \text {FresnelC}(b x) \left (4 b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+66 b x \sin \left (b^2 \pi x^2\right )}{120 b^5 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 209, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\FresnelC \left (b x \right )^{2} b^{5} x^{5}}{5}-2 \FresnelC \left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{5 \pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}+\frac {\frac {2 b x \sin \left (b^{2} \pi \,x^{2}\right )}{5 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{5 \pi }}{\pi ^{2}}+\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{5 \pi ^{3}}}{b^{5}}\) | \(209\) |
default | \(\frac {\frac {\FresnelC \left (b x \right )^{2} b^{5} x^{5}}{5}-2 \FresnelC \left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{5 \pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}+\frac {\frac {2 b x \sin \left (b^{2} \pi \,x^{2}\right )}{5 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{5 \pi }}{\pi ^{2}}+\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{5 \pi ^{3}}}{b^{5}}\) | \(209\) |
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 = 0.38, size = 149, normalized size = 0.84 \begin {gather*} \frac {24 \, \pi ^{3} b^{6} x^{5} \operatorname {C}\left (b x\right )^{2} - 24 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 44 \, \pi b^{4} x^{3} - 192 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 129 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 12 \, {\left (11 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, \pi ^{3} b^{6}} \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^{4} C^{2}\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^4\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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