Optimal. Leaf size=27 \[ \frac {1}{2} b \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x)}{x} \]
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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 = {6562, 3457}
\begin {gather*} \frac {1}{2} b \text {CosIntegral}\left (\frac {1}{2} \pi b^2 x^2\right )-\frac {\text {FresnelC}(b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3457
Rule 6562
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 = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {1}{2} b \text {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\text {FresnelC}(b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 28, normalized size = 1.04
method | result | size |
derivativedivides | \(b \left (-\frac {\FresnelC \left (b x \right )}{b x}+\frac {\cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
default | \(b \left (-\frac {\FresnelC \left (b x \right )}{b x}+\frac {\cosineIntegral \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {\pi ^{\frac {3}{2}} x^{4} b^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2, \frac {9}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{10}+\frac {8 \gamma -8 \ln \left (2\right )-16+16 \ln \left (x \right )+8 \ln \left (\pi \right )+16 \ln \left (b \right )}{\sqrt {\pi }}\right )}{16}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.35, size = 34, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b {\left ({\rm Ei}\left (\frac {1}{2} i \, \pi b^{2} x^{2}\right ) + {\rm Ei}\left (-\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} - \frac {\operatorname {C}\left (b x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 1.41 \begin {gather*} \frac {b x \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + b x \operatorname {Ci}\left (-\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, \operatorname {C}\left (b x\right )}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (20) = 40\).
time = 0.50, size = 53, normalized size = 1.96 \begin {gather*} - \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} \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 = -0.04 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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