Integrand size = 14, antiderivative size = 109 \[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\frac {x}{2 b}-\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2} \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6655, 6874, 2715, 8, 3393, 3383, 6647, 4491, 12, 3380} \[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}-\frac {x \operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {x}{2 b} \]
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Rule 8
Rule 12
Rule 2715
Rule 3380
Rule 3383
Rule 3393
Rule 4491
Rule 6647
Rule 6655
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}+\frac {\int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx}{b}+\int \frac {x \cos ^2(a+b x)}{a+b x} \, dx \\ & = -\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b}+\int \left (\frac {\cos ^2(a+b x)}{b}-\frac {a \cos ^2(a+b x)}{b (a+b x)}\right ) \, dx \\ & = -\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}+\frac {\int \cos ^2(a+b x) \, dx}{b}-\frac {\int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b}-\frac {a \int \frac {\cos ^2(a+b x)}{a+b x} \, dx}{b} \\ & = -\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}+\frac {\int 1 \, dx}{2 b}-\frac {\int \frac {\sin (2 a+2 b x)}{a+b x} \, dx}{2 b}-\frac {a \int \left (\frac {1}{2 (a+b x)}+\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b} \\ & = \frac {x}{2 b}-\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}-\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}-\frac {a \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b} \\ & = \frac {x}{2 b}-\frac {x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\frac {2 b x-2 a \operatorname {CosIntegral}(2 (a+b x))-2 a \log (a+b x)+\operatorname {CosIntegral}(a+b x) (-4 b x \cos (a+b x)+4 \sin (a+b x))+\sin (2 (a+b x))-2 \text {Si}(2 (a+b x))}{4 b^2} \]
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Time = 1.49 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-a \left (\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b^{2}}\) | \(96\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-a \left (\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b^{2}}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.51 \[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=-\frac {2 \, \pi b^{2} x \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) - 2 \, \pi b \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + \sqrt {b^{2}} {\left ({\left (\pi a + 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) - \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (\pi a - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (\pi a + 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (\pi a - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \]
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\[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\int x \sin {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
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\[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\int { x \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) \,d x } \]
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\[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\int { x \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \operatorname {CosIntegral}(a+b x) \sin (a+b x) \, dx=\int x\,\mathrm {cosint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]
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