Integrand size = 12, antiderivative size = 44 \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2 \]
[Out]
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6656, 6818, 12, 4491, 3378, 3383} \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(2 b x)-\frac {1}{2} b \text {Si}(b x)^2-\frac {\text {Si}(b x) \cos (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]
[In]
[Out]
Rule 12
Rule 3378
Rule 3383
Rule 4491
Rule 6656
Rule 6818
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (b x) \text {Si}(b x)}{x}+b \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx-b \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx \\ & = -\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\int \frac {\cos (b x) \sin (b x)}{x^2} \, dx \\ & = -\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\int \frac {\sin (2 b x)}{2 x^2} \, dx \\ & = -\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2} \, dx \\ & = -\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+b \int \frac {\cos (2 b x)}{x} \, dx \\ & = b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2 \\ \end{align*}
\[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx \]
[In]
[Out]
\[\int \frac {\cos \left (b x \right ) \operatorname {Si}\left (b x \right )}{x^{2}}d x\]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=-\frac {b x \operatorname {Si}\left (b x\right )^{2} - 2 \, b x \operatorname {Ci}\left (2 \, b x\right ) + 2 \, \cos \left (b x\right ) \sin \left (b x\right ) + 2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{2 \, x} \]
[In]
[Out]
\[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int \frac {\cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}}{x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int \frac {\mathrm {sinint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^2} \,d x \]
[In]
[Out]