Integrand size = 8, antiderivative size = 49 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)+\frac {2 \sin (b x)}{3 b^3}-\frac {x^2 \sin (b x)}{3 b} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3377, 2717} \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {2 \sin (b x)}{3 b^3}-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b} \]
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Rule 12
Rule 2717
Rule 3377
Rule 6639
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {1}{3} b \int \frac {x^2 \cos (b x)}{b} \, dx \\ & = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {1}{3} \int x^2 \cos (b x) \, dx \\ & = \frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int x \sin (b x) \, dx}{3 b} \\ & = -\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {x^2 \sin (b x)}{3 b}+\frac {2 \int \cos (b x) \, dx}{3 b^2} \\ & = -\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)+\frac {2 \sin (b x)}{3 b^3}-\frac {x^2 \sin (b x)}{3 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=-\frac {2 x \cos (b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)-\frac {\left (-2+b^2 x^2\right ) \sin (b x)}{3 b^3} \]
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Time = 0.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )}{3 b^{3}}\) | \(42\) |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )}{3}+\frac {2 \sin \left (b x \right )}{3}-\frac {2 b x \cos \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sin \left (b x \right )}{3}+\frac {2 \sin \left (b x \right )}{3}-\frac {2 b x \cos \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
meijerg | \(\frac {2 \sqrt {\pi }\, \left (-\frac {b^{5} x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {7}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{40 \sqrt {\pi }}+\frac {\left (-\frac {2}{3}+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{3} b^{3}}{12 \sqrt {\pi }}\right )}{b^{3}}\) | \(63\) |
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Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - \pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
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Time = 1.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=- \frac {x^{3} \log {\left (b x \right )}}{3} + \frac {x^{3} \log {\left (b^{2} x^{2} \right )}}{6} + \frac {x^{3} \operatorname {Ci}{\left (b x \right )}}{3} - \frac {x^{2} \sin {\left (b x \right )}}{3 b} - \frac {2 x \cos {\left (b x \right )}}{3 b^{2}} + \frac {2 \sin {\left (b x \right )}}{3 b^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x\right ) - \frac {\pi b^{2} x^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{3 \, \pi ^{2} b^{3}} \]
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\[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\int { x^{2} \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^2 \operatorname {CosIntegral}(b x) \, dx=\frac {x^3\,\mathrm {cosint}\left (b\,x\right )}{3}-\frac {b^2\,x^2\,\sin \left (b\,x\right )-2\,\sin \left (b\,x\right )+2\,b\,x\,\cos \left (b\,x\right )}{3\,b^3} \]
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