Integrand size = 14, antiderivative size = 96 \[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \]
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Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6649, 4669, 6873, 6874, 2718, 3380, 6653, 3393, 3383} \[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=-\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b^2}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b} \]
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Rule 2718
Rule 3380
Rule 3383
Rule 3393
Rule 4669
Rule 6649
Rule 6653
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\int \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx}{b}-\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx \\ & = \frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x} \, dx-\frac {\int \frac {\cos ^2(a+b x)}{a+b x} \, dx}{b} \\ & = \frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x} \, dx-\frac {\int \left (\frac {1}{2 (a+b x)}+\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b} \\ & = \frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx-\frac {\int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b} \\ & = \frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\int \sin (2 a+2 b x) \, dx}{2 b}-\frac {a \int \frac {\sin (2 a+2 b x)}{-a-b x} \, dx}{2 b} \\ & = \frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\cos (2 (a+b x))-2 \operatorname {CosIntegral}(2 (a+b x))-2 \log (a+b x)+4 \operatorname {CosIntegral}(a+b x) (\cos (a+b x)+b x \sin (a+b x))+2 a \text {Si}(2 (a+b x))}{4 b^2} \]
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Time = 2.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}+\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}+\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (88) = 176\).
Time = 0.29 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.88 \[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {2 \, \pi b^{2} x \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, \pi b \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) - 2 \, b \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) \sin \left (b x + a\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (\pi a + 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (\pi a - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (\pi a + 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) - \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left ({\left (\pi a - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \]
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\[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int x \cos {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
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\[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) \,d x } \]
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\[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int x\,\mathrm {cosint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]
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