Optimal. Leaf size=94 \[ -\frac {1}{90} \pi ^3 b^6 C(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{30 x^5}+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x^3}-\frac {S(b x)}{6 x^6} \]
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6426, 3387, 3388, 3352} \[ -\frac {1}{90} \pi ^3 b^6 \text {FresnelC}(b x)+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x}-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{30 x^5}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x^3}-\frac {S(b x)}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 3352
Rule 3387
Rule 3388
Rule 6426
Rubi steps
\begin {align*} \int \frac {S(b x)}{x^7} \, dx &=-\frac {S(b x)}{6 x^6}+\frac {1}{6} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac {S(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {S(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {1}{90} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {S(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {1}{90} \left (b^7 \pi ^3\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {1}{90} b^6 \pi ^3 C(b x)-\frac {S(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 0.81 \[ \frac {1}{90} \left (-\pi ^3 b^6 C(b x)+\frac {b \left (\pi ^2 b^4 x^4-3\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^5}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^3}-\frac {15 S(b x)}{x^6}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 86, normalized size = 0.91 \[ b^{6} \left (-\frac {\mathrm {S}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \FresnelC \left (b x \right )\right )}{3}\right )}{30}\right ) \]
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 \[ \int \frac {{\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]
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 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 56, normalized size = 0.60 \[ \frac {\pi b^{3} \Gamma \left (- \frac {3}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4}, \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 x^{3} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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