Integrand size = 8, antiderivative size = 26 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=-\frac {\cos (b x)}{x}-\frac {\operatorname {CosIntegral}(b x)}{x}-b \text {Si}(b x) \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3378, 3380} \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=-\frac {\operatorname {CosIntegral}(b x)}{x}-b \text {Si}(b x)-\frac {\cos (b x)}{x} \]
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Rule 12
Rule 3378
Rule 3380
Rule 6639
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {CosIntegral}(b x)}{x}+b \int \frac {\cos (b x)}{b x^2} \, dx \\ & = -\frac {\operatorname {CosIntegral}(b x)}{x}+\int \frac {\cos (b x)}{x^2} \, dx \\ & = -\frac {\cos (b x)}{x}-\frac {\operatorname {CosIntegral}(b x)}{x}-b \int \frac {\sin (b x)}{x} \, dx \\ & = -\frac {\cos (b x)}{x}-\frac {\operatorname {CosIntegral}(b x)}{x}-b \text {Si}(b x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=-\frac {\cos (b x)}{x}-\frac {\operatorname {CosIntegral}(b x)}{x}-b \text {Si}(b x) \]
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Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
parts | \(-\frac {\operatorname {Ci}\left (b x \right )}{x}+b \left (-\frac {\cos \left (b x \right )}{b x}-\operatorname {Si}\left (b x \right )\right )\) | \(32\) |
derivativedivides | \(b \left (-\frac {\operatorname {Ci}\left (b x \right )}{b x}-\frac {\cos \left (b x \right )}{b x}-\operatorname {Si}\left (b x \right )\right )\) | \(34\) |
default | \(b \left (-\frac {\operatorname {Ci}\left (b x \right )}{b x}-\frac {\cos \left (b x \right )}{b x}-\operatorname {Si}\left (b x \right )\right )\) | \(34\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {2 b x \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1, 1\right ], \left [\frac {3}{2}, \frac {3}{2}, 2, 2\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }}-\frac {4 \left (2+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right )}{\sqrt {\pi }\, x b}\right )}{8}\) | \(57\) |
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none
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=\frac {b x \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \operatorname {C}\left (b x\right )}{2 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=- \frac {b^{2} x {{}_{3}F_{4}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ \frac {3}{2}, \frac {3}{2}, 2, 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{4} - \frac {\log {\left (b^{2} x^{2} \right )}}{2 x} - \frac {1}{x} - \frac {\gamma }{x} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=\frac {1}{4} \, b {\left ({\rm Ei}\left (\frac {1}{2} i \, \pi b^{2} x^{2}\right ) + {\rm Ei}\left (-\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} - \frac {\operatorname {C}\left (b x\right )}{x} \]
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\[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x)}{x^2} \, dx=-b\,\mathrm {sinint}\left (b\,x\right )-\frac {\mathrm {cosint}\left (b\,x\right )}{x}-\frac {\cos \left (b\,x\right )}{x} \]
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