Integrand size = 12, antiderivative size = 12 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \text {Int}\left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x},x\right ) \]
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Not integrable
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2} \, dx+\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{b x^3} \, dx \\ & = -\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\cos (b x) \sin (b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\cos ^2(b x)}{b x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ & = -\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin (2 b x)}{2 x^3} \, dx+\frac {1}{2} b \int \frac {\cos ^2(b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx+b^2 \int -\frac {\sin (2 b x)}{2 x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}+\frac {1}{4} b \int \frac {\cos (2 b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-\frac {1}{2} b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 1.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )}{x^{3}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 2.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
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Not integrable
Time = 5.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \]
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