Integrand size = 13, antiderivative size = 33 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6647, 4491, 12, 3380} \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \]
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Rule 12
Rule 3380
Rule 4491
Rule 6647
Rubi steps \begin{align*} \text {integral}& = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x} \, dx \\ & = \frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 (a+b x))}{2 b} \]
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Time = 1.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x +a \right ) \sin \left (b x +a \right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}}{b}\) | \(30\) |
default | \(\frac {\operatorname {Ci}\left (b x +a \right ) \sin \left (b x +a \right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}}{b}\) | \(30\) |
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Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.82 \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {2 \, b \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) - \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \cos \left (\frac {1}{2 \, \pi }\right ) \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right ) - \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) \sin \left (\frac {1}{2 \, \pi }\right )}{2 \, b^{2}} \]
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\[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int \cos {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
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\[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int { \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) \,d x } \]
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\[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int { \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\int \mathrm {cosint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]
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