Optimal. Leaf size=216 \[ \frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {531 \text {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {48 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6589, 6597,
3472, 30, 3467, 3466, 3433, 6595, 3438} \begin {gather*} -\frac {531 \text {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} \pi ^4 b^8}+\frac {24 x}{\pi ^4 b^7}-\frac {3 x^5}{5 \pi ^2 b^3}-\frac {x^6 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {48 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac {147 x \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^7}+\frac {24 x^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {17 x^3 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x^5 \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 3433
Rule 3438
Rule 3466
Rule 3467
Rule 3472
Rule 6589
Rule 6595
Rule 6597
Rubi steps
\begin {align*} \int x^7 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx &=-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {6 \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^2 \pi }+\frac {\int x^6 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {24 \int x^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}+\frac {5 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {6 \int x^4 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {5 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {48 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^6 \pi ^3}-\frac {15 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}-\frac {12 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {3 \int x^4 \, dx}{b^3 \pi ^2}+\frac {3 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {111 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {48 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 \int \cos \left (b^2 \pi x^2\right ) \, dx}{16 b^7 \pi ^4}-\frac {6 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}+\frac {48 \int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}-\frac {9 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}\\ &=-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {15 C\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {3 \sqrt {2} C\left (\sqrt {2} b x\right )}{b^8 \pi ^4}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {48 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {9 \int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}+\frac {48 \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^7 \pi ^4}\\ &=\frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {51 C\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {3 \sqrt {2} C\left (\sqrt {2} b x\right )}{b^8 \pi ^4}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {48 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {24 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}\\ &=\frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}+\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}-\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {51 C\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {15 \sqrt {2} C\left (\sqrt {2} b x\right )}{b^8 \pi ^4}+\frac {24 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac {x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac {48 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac {6 x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 153, normalized size = 0.71 \begin {gather*} \frac {-2655 \sqrt {2} \text {FresnelC}\left (\sqrt {2} b x\right )-160 S(b x) \left (b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-6 \left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (\left (735-20 b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )+2 \left (960-24 b^4 \pi ^2 x^4+85 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )\right )}{160 b^8 \pi ^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 318, normalized size = 1.47
method | result | size |
default | \(\frac {\frac {\mathrm {S}\left (b x \right ) \left (-\frac {b^{6} x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {6 b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {24 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {3}{5} \pi ^{2} b^{5} x^{5}-24 b x}{\pi ^{4}}-\frac {3 \left (\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )\right )}{\pi ^{4}}-\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \FresnelC \left (b x \sqrt {2}\right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\) | \(318\) |
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 = 167, normalized size = 0.77 \begin {gather*} -\frac {56 \, \pi ^{2} b^{6} x^{5} - 2370 \, b^{2} x + 20 \, {\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 160 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2655 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 40 \, {\left (17 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 24 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{160 \, \pi ^{4} b^{9}} \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^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\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 = -0.00 \begin {gather*} \int x^7\,\mathrm {FresnelS}\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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