Integrand size = 17, antiderivative size = 15 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} (-\cos (x)+\sin (x))^{2/3} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3224} \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} (\sin (x)-\cos (x))^{2/3} \]
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Rule 3224
Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} (-\cos (x)+\sin (x))^{2/3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} (-\cos (x)+\sin (x))^{2/3} \]
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Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {3 \left (-\cos \left (x \right )+\sin \left (x \right )\right )^{\frac {2}{3}}}{2}\) | \(12\) |
default | \(\frac {3 \left (-\cos \left (x \right )+\sin \left (x \right )\right )^{\frac {2}{3}}}{2}\) | \(12\) |
risch | \(\frac {\left (-\frac {3}{2}-\frac {3 i}{2}\right ) {\left (\left (1+i\right ) \left (-{\mathrm e}^{4 i x}+i {\mathrm e}^{2 i x}\right )\right )}^{\frac {1}{3}} \left ({\mathrm e}^{i x}-i {\mathrm e}^{-i x}\right )}{\left (-8 \cos \left (x \right )+8 \sin \left (x \right )\right )^{\frac {1}{3}} {\left (\left (-1-i\right ) \left ({\mathrm e}^{4 i x}-i {\mathrm e}^{2 i x}\right )\right )}^{\frac {1}{3}}}\) | \(72\) |
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none
Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \]
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Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3 \left (\sin {\left (x \right )} - \cos {\left (x \right )}\right )^{\frac {2}{3}}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \]
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Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx=\frac {3\,2^{1/3}\,{\left (-\cos \left (x+\frac {\pi }{4}\right )\right )}^{2/3}}{2} \]
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