Optimal. Leaf size=253 \[ -\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3} \]
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Rubi [A]
time = 0.28, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6565, 6589,
6597, 3460, 3390, 30, 3377, 2717, 2714, 6575} \begin {gather*} -\frac {105 S(b x)^2}{8 \pi ^4 b^8}-\frac {105 x^2}{16 \pi ^4 b^6}+\frac {x^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}+\frac {7 x^6}{48 \pi ^2 b^2}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{16 \pi ^2 b^2}+\frac {10 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^8}+\frac {105 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^6}-\frac {35 x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^3 b^5}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{8} x^8 S(b x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2714
Rule 2717
Rule 3377
Rule 3390
Rule 3460
Rule 6565
Rule 6575
Rule 6589
Rule 6597
Rubi steps
\begin {align*} \int x^7 S(b x)^2 \, dx &=\frac {1}{8} x^8 S(b x)^2-\frac {1}{4} b \int x^8 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{8 \pi }-\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{4 b \pi }\\ &=\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {35 \int x^4 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}+\frac {7 \int x^5 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^2 \pi ^2}-\frac {\text {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 \pi }\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {105 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{4 b^5 \pi ^3}+\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^4 \pi ^3}-\frac {3 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}+\frac {7 \text {Subst}\left (\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}\\ &=\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{16 b^4 \pi ^3}-\frac {105 \int S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}-\frac {105 \int x \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^6 \pi ^4}+\frac {3 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}+\frac {35 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^4 \pi ^3}+\frac {7 \text {Subst}\left (\int x^2 \, dx,x,x^2\right )}{16 b^2 \pi ^2}-\frac {7 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^2 \pi ^2}\\ &=\frac {7 x^6}{48 b^2 \pi ^2}-\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac {105 \text {Subst}(\int x \, dx,x,S(b x))}{4 b^8 \pi ^4}+\frac {3 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}+\frac {35 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{16 b^6 \pi ^4}-\frac {105 \text {Subst}\left (\int \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}+\frac {7 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^4 \pi ^3}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {73 \sin \left (b^2 \pi x^2\right )}{8 b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac {7 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^6 \pi ^4}\\ &=-\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 253, normalized size = 1.00 \begin {gather*} -\frac {105 x^2}{16 b^6 \pi ^4}+\frac {7 x^6}{48 b^2 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{16 b^6 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{16 b^2 \pi ^2}-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{4 b \pi }-\frac {105 S(b x)^2}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)^2+\frac {105 x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac {7 x^5 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {10 \sin \left (b^2 \pi x^2\right )}{b^8 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int x^{7} \mathrm {S}\left (b x \right )^{2}\, dx\]
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 = 0.35, size = 183, normalized size = 0.72 \begin {gather*} \frac {2 \, \pi ^{3} b^{6} x^{6} - 75 \, \pi b^{2} x^{2} + 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 3 \, {\left (105 \, \pi - \pi ^{5} b^{8} x^{8}\right )} \operatorname {S}\left (b x\right )^{2} - 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 7 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{5} b^{8}} \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} S^{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 = -0.00 \begin {gather*} \int x^7\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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