Integrand size = 9, antiderivative size = 34 \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {\operatorname {CosIntegral}(2 b x)}{2 b}-\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b} \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6652, 12, 3393, 3383} \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {\operatorname {CosIntegral}(2 b x)}{2 b}+\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\log (x)}{2 b} \]
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Rule 12
Rule 3383
Rule 3393
Rule 6652
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (b x) \text {Si}(b x)}{b}-\int \frac {\sin ^2(b x)}{b x} \, dx \\ & = \frac {\sin (b x) \text {Si}(b x)}{b}-\frac {\int \frac {\sin ^2(b x)}{x} \, dx}{b} \\ & = \frac {\sin (b x) \text {Si}(b x)}{b}-\frac {\int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b} \\ & = -\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b}+\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b} \\ & = \frac {\operatorname {CosIntegral}(2 b x)}{2 b}-\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {\operatorname {CosIntegral}(2 b x)}{2 b}-\frac {\log (b x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b} \]
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Time = 0.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (b x \right ) \sin \left (b x \right )-\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Ci}\left (2 b x \right )}{2}}{b}\) | \(28\) |
default | \(\frac {\operatorname {Si}\left (b x \right ) \sin \left (b x \right )-\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Ci}\left (2 b x \right )}{2}}{b}\) | \(28\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {2 \, \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) + \operatorname {Ci}\left (2 \, b x\right ) - \log \left (x\right )}{2 \, b} \]
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\[ \int \cos (b x) \text {Si}(b x) \, dx=\int \cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \]
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\[ \int \cos (b x) \text {Si}(b x) \, dx=\int { \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {\sin \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} + \frac {\operatorname {Ci}\left (2 \, b x\right ) + \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \left (x\right )}{4 \, b} \]
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Timed out. \[ \int \cos (b x) \text {Si}(b x) \, dx=\frac {\mathrm {cosint}\left (2\,b\,x\right )-\ln \left (x\right )+2\,\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right )}{2\,b} \]
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