Integrand size = 10, antiderivative size = 155 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 a+2 b x)}{4 b^2}-\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {a \text {Si}(2 a+2 b x)}{b^2} \]
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Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6645, 6649, 4669, 6873, 6874, 2718, 3380, 6653, 3393, 3383, 6641, 6647, 4491, 12} \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {\operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{b^2}+\frac {\log (a+b x)}{2 b^2}-\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b} \]
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Rule 12
Rule 2718
Rule 3380
Rule 3383
Rule 3393
Rule 4491
Rule 4669
Rule 6641
Rule 6645
Rule 6647
Rule 6649
Rule 6653
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\frac {a \int \operatorname {CosIntegral}(a+b x)^2 \, dx}{2 b}-\int x \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx \\ & = -\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}-\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}+\frac {a \int \cos (a+b x) \operatorname {CosIntegral}(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 {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (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 {a \int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b} \\ & = -\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (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 {a \int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b} \\ & = -\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{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 {a \int \frac {\sin (2 a+2 b x)}{a+b x} \, dx}{2 b} \\ & = -\frac {\cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}-\frac {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{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}+\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 {a (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \operatorname {CosIntegral}(a+b x)^2}{2 b}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^2}-\frac {x \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {a \text {Si}(2 a+2 b x)}{b^2} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=-\frac {\cos (2 (a+b x))+2 \left (a^2-b^2 x^2\right ) \operatorname {CosIntegral}(a+b x)^2-2 \operatorname {CosIntegral}(2 (a+b x))-2 \log (a+b x)+4 \operatorname {CosIntegral}(a+b x) (\cos (a+b x)+(-a+b x) \sin (a+b x))+4 a \text {Si}(2 (a+b x))}{4 b^2} \]
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Time = 0.90 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(113\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Ci}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\frac {\cos \left (b x +a \right )}{2}+\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}\right )-a \,\operatorname {Si}\left (2 b x +2 a \right )-\frac {\cos \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(113\) |
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\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x \operatorname {Ci}^{2}{\left (a + b x \right )}\, dx \]
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\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int { x \operatorname {C}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x \operatorname {CosIntegral}(a+b x)^2 \, dx=\int x\,{\mathrm {cosint}\left (a+b\,x\right )}^2 \,d x \]
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