Integrand size = 10, antiderivative size = 60 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2} \]
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Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6649, 12, 2644, 30, 6653, 3393, 3383} \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b^2}-\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b} \]
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Rule 12
Rule 30
Rule 2644
Rule 3383
Rule 3393
Rule 6649
Rule 6653
Rubi steps \begin{align*} \text {integral}& = \frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x) \, dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b} \, dx \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{b x} \, dx}{b}-\frac {\int \cos (b x) \sin (b x) \, dx}{b} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b^2}-\frac {\text {Subst}(\int x \, dx,x,\sin (b x))}{b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b^2} \\ & = \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {\cos (2 b x)-2 \operatorname {CosIntegral}(2 b x)-2 \log (x)+4 \operatorname {CosIntegral}(b x) (\cos (b x)+b x \sin (b x))}{4 b^2} \]
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Time = 0.77 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}+\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\) | \(44\) |
default | \(\frac {\operatorname {Ci}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}+\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (54) = 108\).
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.68 \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {2 \, \pi b^{2} x \operatorname {C}\left (b x\right ) \sin \left (b x\right ) + 2 \, \pi b \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - 2 \, b \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \sin \left (b x\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \]
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\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
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\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \]
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\[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x\,\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]
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