Optimal. Leaf size=44 \[ -\frac {1}{2} \pi b^2 S(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x}-\frac {C(b x)}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6427, 3388, 3351} \[ -\frac {1}{2} \pi b^2 S(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x}-\frac {\text {FresnelC}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3388
Rule 6427
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^3} \, dx &=-\frac {C(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{2 x}-\frac {C(b x)}{2 x^2}-\frac {1}{2} \left (b^3 \pi \right ) \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{2 x}-\frac {C(b x)}{2 x^2}-\frac {1}{2} b^2 \pi S(b x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.00 \[ -\frac {1}{2} \pi b^2 S(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x}-\frac {C(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.98 \[ b^{2} \left (-\frac {\FresnelC \left (b x \right )}{2 b^{2} x^{2}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b x}-\frac {\pi \,\mathrm {S}\left (b x \right )}{2}\right ) \]
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 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 51, normalized size = 1.16 \[ \frac {b \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {3}{4}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 x \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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