Integrand size = 7, antiderivative size = 14 \[ \int \sin ^2(x) \tan (x) \, dx=\frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2670, 14} \[ \int \sin ^2(x) \tan (x) \, dx=\frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
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Rule 14
Rule 2670
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\cos (x)\right ) \\ & = \frac {\cos ^2(x)}{2}-\log (\cos (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sin ^2(x) \tan (x) \, dx=\frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
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Time = 1.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {\left (\sin ^{2}\left (x \right )\right )}{2}-\ln \left (\cos \left (x \right )\right )\) | \(13\) |
risch | \(i x +\frac {{\mathrm e}^{2 i x}}{8}+\frac {{\mathrm e}^{-2 i x}}{8}-\ln \left ({\mathrm e}^{2 i x}+1\right )\) | \(30\) |
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none
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sin ^2(x) \tan (x) \, dx=\frac {1}{2} \, \cos \left (x\right )^{2} - \log \left (-\cos \left (x\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sin ^2(x) \tan (x) \, dx=- \log {\left (\cos {\left (x \right )} \right )} + \frac {\cos ^{2}{\left (x \right )}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \sin ^2(x) \tan (x) \, dx=-\frac {1}{2} \, \sin \left (x\right )^{2} - \frac {1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \sin ^2(x) \tan (x) \, dx=\frac {1}{2} \, \cos \left (x\right )^{2} - \log \left ({\left | \cos \left (x\right ) \right |}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \sin ^2(x) \tan (x) \, dx=\frac {{\cos \left (x\right )}^2}{2}+\frac {\ln \left ({\mathrm {tan}\left (x\right )}^2+1\right )}{2} \]
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