Optimal. Leaf size=167 \[ -\frac {4 x}{b^5 \pi ^3}+\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^2 \pi }-\frac {43 \text {FresnelC}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {4 x^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6598, 6590,
6596, 3439, 3433, 3466, 3473, 30, 3467} \begin {gather*} -\frac {43 \text {FresnelC}\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}-\frac {4 x}{\pi ^3 b^5}-\frac {x^4 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {8 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {11 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {4 x^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^3 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {x^5}{10 \pi b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3433
Rule 3439
Rule 3466
Rule 3467
Rule 3473
Rule 6590
Rule 6596
Rule 6598
Rubi steps
\begin {align*} \int x^5 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac {\int x^4 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {4 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {8 \int x C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac {2 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac {\int x^4 \, dx}{2 b \pi }+\frac {\int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {x^5}{10 b \pi }+\frac {x \cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac {4 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {8 \int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {3 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {C\left (\sqrt {2} b x\right )}{\sqrt {2} b^6 \pi ^3}+\frac {4 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {3 \int \cos \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}-\frac {8 \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^5 \pi ^3}\\ &=-\frac {4 x}{b^5 \pi ^3}+\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {11 C\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {4 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {4 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=-\frac {4 x}{b^5 \pi ^3}+\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^6 \pi ^3}-\frac {x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }-\frac {11 C\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}-\frac {2 \sqrt {2} C\left (\sqrt {2} b x\right )}{b^6 \pi ^3}+\frac {4 x^2 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 126, normalized size = 0.75 \begin {gather*} \frac {-215 \sqrt {2} \text {FresnelC}\left (\sqrt {2} b x\right )-80 \text {FresnelC}(b x) \left (\left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-4 b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (-160+4 b^4 \pi ^2 x^4+55 \cos \left (b^2 \pi x^2\right )+10 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )}{80 b^6 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 212, normalized size = 1.27
method | result | size |
default | \(\frac {\frac {\FresnelC \left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{b^{5}}-\frac {-\frac {\frac {1}{5} \pi ^{2} b^{5} x^{5}-8 b x}{2 \pi ^{3}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi }}{\pi ^{2}}-\frac {\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{2 \pi ^{3}}}{b^{5}}}{b}\) | \(212\) |
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 = 139, normalized size = 0.83 \begin {gather*} \frac {8 \, \pi ^{2} b^{6} x^{5} + 220 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 430 \, b^{2} x - 80 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 215 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 40 \, {\left (\pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 8 \, \pi b^{3} x^{2} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{80 \, \pi ^{3} b^{7}} \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^{5} \sin {\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^5\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \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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