Optimal. Leaf size=26 \[ -\frac {\cos (b x)}{x}-\frac {\text {CosIntegral}(b x)}{x}-b \text {Si}(b x) \]
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Rubi [A]
time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3378,
3380} \begin {gather*} -\frac {\text {CosIntegral}(b x)}{x}-b \text {Si}(b x)-\frac {\cos (b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3380
Rule 6639
Rubi steps
\begin {align*} \int \frac {\text {Ci}(b x)}{x^2} \, dx &=-\frac {\text {Ci}(b x)}{x}+b \int \frac {\cos (b x)}{b x^2} \, dx\\ &=-\frac {\text {Ci}(b x)}{x}+\int \frac {\cos (b x)}{x^2} \, dx\\ &=-\frac {\cos (b x)}{x}-\frac {\text {Ci}(b x)}{x}-b \int \frac {\sin (b x)}{x} \, dx\\ &=-\frac {\cos (b x)}{x}-\frac {\text {Ci}(b x)}{x}-b \text {Si}(b x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\cos (b x)}{x}-\frac {\text {CosIntegral}(b x)}{x}-b \text {Si}(b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 34, normalized size = 1.31
method | result | size |
derivativedivides | \(b \left (-\frac {\cosineIntegral \left (b x \right )}{b x}-\frac {\cos \left (b x \right )}{b x}-\sinIntegral \left (b x \right )\right )\) | \(34\) |
default | \(b \left (-\frac {\cosineIntegral \left (b x \right )}{b x}-\frac {\cos \left (b x \right )}{b x}-\sinIntegral \left (b x \right )\right )\) | \(34\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {2 b x \hypergeom \left (\left [\frac {1}{2}, 1, 1\right ], \left [\frac {3}{2}, \frac {3}{2}, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }}-\frac {4 \left (2+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right )}{\sqrt {\pi }\, x b}\right )}{8}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.31, size = 34, normalized size = 1.31 \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 = 1.46 \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. 42 vs.
\(2 (20) = 40\).
time = 0.53, size = 42, normalized size = 1.62 \begin {gather*} - \frac {b^{2} x {{}_{3}F_{4}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ \frac {3}{2}, \frac {3}{2}, 2, 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{4} - \frac {\log {\left (b^{2} x^{2} \right )}}{2 x} - \frac {1}{x} - \frac {\gamma }{x} \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*} -b\,\mathrm {sinint}\left (b\,x\right )-\frac {\mathrm {cosint}\left (b\,x\right )}{x}-\frac {\cos \left (b\,x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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