Optimal. Leaf size=27 \[ \frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6427, 3376} \[ \frac {1}{2} b \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )-\frac {\text {FresnelC}(b x)}{x} \]
Antiderivative was successfully verified.
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Rule 3376
Rule 6427
Rubi steps
\begin {align*} \int \frac {C(b x)}{x^2} \, dx &=-\frac {C(b x)}{x}+b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=\frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.00 \[ \frac {1}{2} b \text {Ci}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {C(b x)}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.04 \[ b \left (-\frac {\FresnelC \left (b x \right )}{b x}+\frac {\Ci \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right ) \]
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 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.06, size = 53, normalized size = 1.96 \[ - \frac {\pi ^{2} b^{5} x^{4} \Gamma \left (\frac {5}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2, \frac {9}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{128 \Gamma \left (\frac {9}{4}\right )} + \frac {b \log {\left (b^{4} x^{4} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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