Optimal. Leaf size=124 \[ -\frac {105 S(b x)}{8 \pi ^4 b^8}+\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac {105 x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}-\frac {35 x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac {7 x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac {1}{8} x^8 S(b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3385, 3386, 3351} \[ -\frac {105 S(b x)}{8 \pi ^4 b^8}-\frac {7 x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac {105 x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac {x^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi b}-\frac {35 x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {1}{8} x^8 S(b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3351
Rule 3385
Rule 3386
Rule 6426
Rubi steps
\begin {align*} \int x^7 S(b x) \, dx &=\frac {1}{8} x^8 S(b x)-\frac {1}{8} b \int x^8 \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 )}{8 b \pi }+\frac {1}{8} x^8 S(b x)-\frac {7 \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b \pi }\\ &=\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {1}{8} x^8 S(b x)-\frac {7 x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {35 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^3 \pi ^2}\\ &=-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {1}{8} x^8 S(b x)-\frac {7 x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac {105 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}\\ &=-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac {1}{8} x^8 S(b x)+\frac {105 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac {105 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{8 b^7 \pi ^4}\\ &=-\frac {35 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac {105 S(b x)}{8 b^8 \pi ^4}+\frac {1}{8} x^8 S(b x)+\frac {105 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {7 x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 88, normalized size = 0.71 \[ \frac {\left (\pi ^4 b^8 x^8-105\right ) S(b x)-7 b x \left (\pi ^2 b^4 x^4-15\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )+\pi b^3 x^3 \left (\pi ^2 b^4 x^4-35\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{7} {\rm fresnels}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{7} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 123, normalized size = 0.99 \[ \frac {\frac {\mathrm {S}\left (b x \right ) b^{8} x^{8}}{8}+\frac {b^{7} x^{7} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 \pi }-\frac {7 \left (\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {5 \left (-\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\mathrm {S}\left (b x \right )}{\pi }}{\pi }\right )}{\pi }\right )}{8 \pi }}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{7} {\rm fresnels}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^7\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.98, size = 184, normalized size = 1.48 \[ \frac {231 x^{8} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{512 \Gamma \left (\frac {15}{4}\right )} + \frac {231 x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi b \Gamma \left (\frac {15}{4}\right )} - \frac {1617 x^{5} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac {15}{4}\right )} - \frac {8085 x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac {15}{4}\right )} + \frac {24255 x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac {15}{4}\right )} - \frac {24255 S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________