Integrand size = 10, antiderivative size = 16 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=-6 x+6 \tan (x)-2 \sin ^2(x) \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {12, 2671, 294, 327, 209} \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=-6 x+6 \tan (x)-2 \sin ^2(x) \tan (x) \]
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Rule 12
Rule 209
Rule 294
Rule 327
Rule 2671
Rubi steps \begin{align*} \text {integral}& = 4 \int \sin ^2(x) \tan ^2(x) \, dx \\ & = 4 \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right ) \\ & = -2 \sin ^2(x) \tan (x)+6 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = 6 \tan (x)-2 \sin ^2(x) \tan (x)-6 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = -6 x+6 \tan (x)-2 \sin ^2(x) \tan (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=4 \left (-\frac {3 x}{2}+\frac {1}{4} \sin (2 x)+\tan (x)\right ) \]
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Time = 2.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {4 \sin \left (x \right )^{5}}{\cos \left (x \right )}+4 \left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )-6 x\) | \(28\) |
risch | \(-6 x -\frac {i {\mathrm e}^{2 i x}}{2}+\frac {i {\mathrm e}^{-2 i x}}{2}+\frac {8 i}{{\mathrm e}^{2 i x}+1}\) | \(33\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=-\frac {2 \, {\left (3 \, x \cos \left (x\right ) - {\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )\right )}}{\cos \left (x\right )} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=- 6 x + \frac {4 \sin ^{3}{\left (x \right )}}{\cos {\left (x \right )}} + 6 \sin {\left (x \right )} \cos {\left (x \right )} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=-6 \, x + \frac {2 \, \tan \left (x\right )}{\tan \left (x\right )^{2} + 1} + 4 \, \tan \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=-6 \, x + \frac {2 \, \tan \left (x\right )}{\tan \left (x\right )^{2} + 1} + 4 \, \tan \left (x\right ) \]
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Time = 26.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int 4 \sin ^2(x) \tan ^2(x) \, dx=2\,\cos \left (x\right )\,\sin \left (x\right )-6\,x+\frac {4\,\sin \left (x\right )}{\cos \left (x\right )} \]
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