Integrand size = 12, antiderivative size = 96 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=b^2 \operatorname {CosIntegral}(2 b x)-\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \sin (2 b x)}{4 x}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2 \]
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Time = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6650, 6656, 6818, 12, 4491, 3378, 3383, 3395, 29, 3393} \[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=b^2 \operatorname {CosIntegral}(2 b x)-\frac {1}{4} b^2 \text {Si}(b x)^2-\frac {\text {Si}(b x) \sin (b x)}{2 x^2}-\frac {b \text {Si}(b x) \cos (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \sin (2 b x)}{4 x}-\frac {b \sin (b x) \cos (b x)}{2 x} \]
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Rule 12
Rule 29
Rule 3378
Rule 3383
Rule 3393
Rule 3395
Rule 4491
Rule 6650
Rule 6656
Rule 6818
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (b x) \text {Si}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\sin ^2(b x)}{b x^3} \, dx+\frac {1}{2} b \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx \\ & = -\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin ^2(b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx-\frac {1}{2} b^2 \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx \\ & = -\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2+\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {1}{x} \, dx-b^2 \int \frac {\sin ^2(b x)}{x} \, dx \\ & = \frac {1}{2} b^2 \log (x)-\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2+\frac {1}{2} b \int \frac {\sin (2 b x)}{2 x^2} \, dx-b^2 \int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx \\ & = -\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2+\frac {1}{4} b \int \frac {\sin (2 b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\cos (2 b x)}{x} \, dx \\ & = \frac {1}{2} b^2 \operatorname {CosIntegral}(2 b x)-\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \sin (2 b x)}{4 x}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2+\frac {1}{2} b^2 \int \frac {\cos (2 b x)}{x} \, dx \\ & = b^2 \operatorname {CosIntegral}(2 b x)-\frac {b \cos (b x) \sin (b x)}{2 x}-\frac {\sin ^2(b x)}{4 x^2}-\frac {b \sin (2 b x)}{4 x}-\frac {b \cos (b x) \text {Si}(b x)}{2 x}-\frac {\sin (b x) \text {Si}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Si}(b x)^2 \\ \end{align*}
\[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=\int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx \]
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\[\int \frac {\operatorname {Si}\left (b x \right ) \sin \left (b x \right )}{x^{3}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.75 \[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=-\frac {b^{2} x^{2} \operatorname {Si}\left (b x\right )^{2} - 4 \, b^{2} x^{2} \operatorname {Ci}\left (2 \, b x\right ) + 2 \, b x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - \cos \left (b x\right )^{2} + 2 \, {\left (2 \, b x \cos \left (b x\right ) + \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) + 1}{4 \, x^{2}} \]
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\[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=\int { \frac {\sin \left (b x\right ) \operatorname {Si}\left (b x\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=\int { \frac {\sin \left (b x\right ) \operatorname {Si}\left (b x\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sin (b x) \text {Si}(b x)}{x^3} \, dx=\int \frac {\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \]
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