Integrand size = 16, antiderivative size = 26 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4419, 272, 36, 29, 31} \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (\sin ^{\sqrt {2}-1}(x)+1\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 4419
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (1+x^{-1+\sqrt {2}}\right )} \, dx,x,\sin (x)\right ) \\ & = \left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right ) \\ & = \left (-1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right )+\left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^{-1+\sqrt {2}}(x)\right ) \\ & = \log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\log (\sin (x))-\left (1+\sqrt {2}\right ) \log \left (1+\sin ^{-1+\sqrt {2}}(x)\right ) \]
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Time = 1.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\left (2+\sqrt {2}\right ) \ln \left (\sin \left (x \right )\right )+\left (-1-\sqrt {2}\right ) \ln \left (\sin \left (x \right )+{\mathrm e}^{\sqrt {2}\, \ln \left (\sin \left (x \right )\right )}\right )\) | \(31\) |
default | \(\left (2+\sqrt {2}\right ) \ln \left (\sin \left (x \right )\right )+\left (-1-\sqrt {2}\right ) \ln \left (\sin \left (x \right )+{\mathrm e}^{\sqrt {2}\, \ln \left (\sin \left (x \right )\right )}\right )\) | \(31\) |
parallelrisch | \(\sqrt {2}\, \left (-\ln \left (\frac {\sin \left (x \right )+\sin \left (x \right )^{\sqrt {2}}}{\cos \left (x \right )+1}\right )+\ln \left (\frac {\csc \left (x \right )}{2}-\frac {\cot \left (x \right )}{2}\right )\right )-\ln \left (\frac {4 \sin \left (x \right )+4 \sin \left (x \right )^{\sqrt {2}}}{\cos \left (x \right )+1}\right )+2 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-\ln \left (\frac {1}{\cos \left (x \right )+1}\right )\) | \(78\) |
risch | \(\text {Expression too large to display}\) | \(674\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=-{\left (\sqrt {2} + 1\right )} \log \left (\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )\right ) + {\left (\sqrt {2} + 2\right )} \log \left (\sin \left (x\right )\right ) \]
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Time = 0.44 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=- \frac {\log {\left (\sin {\left (x \right )} + \sin ^{\sqrt {2}}{\left (x \right )} \right )}}{-1 + \sqrt {2}} + \frac {\sqrt {2} \log {\left (\sin {\left (x \right )} \right )}}{-1 + \sqrt {2}} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\frac {\sqrt {2} \log \left (\sin \left (x\right )\right )}{\sqrt {2} - 1} - \frac {\log \left (\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )\right )}{\sqrt {2} - 1} \]
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\[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\int { \frac {\cos \left (x\right )}{\sin \left (x\right )^{\left (\sqrt {2}\right )} + \sin \left (x\right )} \,d x } \]
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Time = 27.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (x)}{\sin (x)+\sin ^{\sqrt {2}}(x)} \, dx=\ln \left (\sin \left (x\right )\right )\,\left (\sqrt {2}+2\right )-\frac {\ln \left (\sin \left (x\right )+{\sin \left (x\right )}^{\sqrt {2}}\right )}{\sqrt {2}-1} \]
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