Integrand size = 8, antiderivative size = 25 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x} \]
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Time = 0.03 (sec) , antiderivative size = 25, 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 = {6638, 12, 3378, 3383} \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(b x)-\frac {\text {Si}(b x)}{x}-\frac {\sin (b x)}{x} \]
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Rule 12
Rule 3378
Rule 3383
Rule 6638
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Si}(b x)}{x}+b \int \frac {\sin (b x)}{b x^2} \, dx \\ & = -\frac {\text {Si}(b x)}{x}+\int \frac {\sin (b x)}{x^2} \, dx \\ & = -\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x}+b \int \frac {\cos (b x)}{x} \, dx \\ & = b \operatorname {CosIntegral}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x} \]
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Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
parts | \(-\frac {\operatorname {Si}\left (b x \right )}{x}+b \left (-\frac {\sin \left (b x \right )}{b x}+\operatorname {Ci}\left (b x \right )\right )\) | \(30\) |
derivativedivides | \(b \left (-\frac {\operatorname {Si}\left (b x \right )}{b x}-\frac {\sin \left (b x \right )}{b x}+\operatorname {Ci}\left (b x \right )\right )\) | \(32\) |
default | \(b \left (-\frac {\operatorname {Si}\left (b x \right )}{b x}-\frac {\sin \left (b x \right )}{b x}+\operatorname {Ci}\left (b x \right )\right )\) | \(32\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {2 b^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {3}{2}\right ], \left [2, 2, \frac {5}{2}, \frac {5}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{9 \sqrt {\pi }}+\frac {8 \gamma -16+8 \ln \left (x \right )+8 \ln \left (b \right )}{\sqrt {\pi }}\right )}{8}\) | \(55\) |
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none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=\frac {b x \operatorname {Ci}\left (b x\right ) - \sin \left (b x\right ) - \operatorname {Si}\left (b x\right )}{x} \]
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Time = 0.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=- \frac {b^{3} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {3}{2} \\ 2, 2, \frac {5}{2}, \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{36} + \frac {b \log {\left (b^{2} x^{2} \right )}}{2} \]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=\frac {1}{2} \, b {\left (\Gamma \left (-1, i \, b x\right ) + \Gamma \left (-1, -i \, b x\right )\right )} - \frac {\operatorname {Si}\left (b x\right )}{x} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=\frac {b x \operatorname {Ci}\left (b x\right ) + b x \operatorname {Ci}\left (-b x\right ) - 2 \, \sin \left (b x\right )}{2 \, x} - \frac {\operatorname {Si}\left (b x\right )}{x} \]
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Timed out. \[ \int \frac {\text {Si}(b x)}{x^2} \, dx=b\,\mathrm {cosint}\left (b\,x\right )-\frac {\mathrm {sinint}\left (b\,x\right )}{x}-\frac {\sin \left (b\,x\right )}{x} \]
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