Integrand size = 8, antiderivative size = 29 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=-\frac {\cos (2 x)}{2 x}-\frac {\sin (2 x)}{4 x^2}-\text {Si}(2 x) \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4491, 12, 3378, 3380} \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=-\text {Si}(2 x)-\frac {\sin (2 x)}{4 x^2}-\frac {\cos (2 x)}{2 x} \]
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Rule 12
Rule 3378
Rule 3380
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin (2 x)}{2 x^3} \, dx \\ & = \frac {1}{2} \int \frac {\sin (2 x)}{x^3} \, dx \\ & = -\frac {\sin (2 x)}{4 x^2}+\frac {1}{2} \int \frac {\cos (2 x)}{x^2} \, dx \\ & = -\frac {\cos (2 x)}{2 x}-\frac {\sin (2 x)}{4 x^2}-\int \frac {\sin (2 x)}{x} \, dx \\ & = -\frac {\cos (2 x)}{2 x}-\frac {\sin (2 x)}{4 x^2}-\text {Si}(2 x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=-\frac {\cos (2 x)}{2 x}-\frac {\sin (2 x)}{4 x^2}-\text {Si}(2 x) \]
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Time = 0.62 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {\cos \left (2 x \right )}{2 x}-\operatorname {Si}\left (2 x \right )-\frac {\sin \left (2 x \right )}{4 x^{2}}\) | \(26\) |
risch | \(\frac {\pi \,\operatorname {csgn}\left (x \right )}{2}-\operatorname {Si}\left (2 x \right )-\frac {\cos \left (2 x \right )}{2 x}-\frac {\sin \left (2 x \right )}{4 x^{2}}\) | \(31\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (-\frac {2 \cos \left (2 x \right )}{x \sqrt {\pi }}-\frac {\sin \left (2 x \right )}{x^{2} \sqrt {\pi }}-\frac {4 \,\operatorname {Si}\left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(40\) |
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none
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=-\frac {2 \, x \cos \left (x\right )^{2} + 2 \, x^{2} \operatorname {Si}\left (2 \, x\right ) + \cos \left (x\right ) \sin \left (x\right ) - x}{2 \, x^{2}} \]
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Time = 0.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=- \operatorname {Si}{\left (2 x \right )} - \frac {\cos {\left (2 x \right )}}{2 x} - \frac {\sin {\left (2 x \right )}}{4 x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=i \, \Gamma \left (-2, 2 i \, x\right ) - i \, \Gamma \left (-2, -2 i \, x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=-\frac {4 \, x^{2} \operatorname {Si}\left (2 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )}{4 \, x^{2}} \]
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Timed out. \[ \int \frac {\cos (x) \sin (x)}{x^3} \, dx=\int \frac {\cos \left (x\right )\,\sin \left (x\right )}{x^3} \,d x \]
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