Integrand size = 17, antiderivative size = 23 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \]
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Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3738, 4210, 2644, 30} \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \]
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Rule 30
Rule 2644
Rule 3738
Rule 4210
Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan ^2(x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx \\ & = \frac {\sec (x) \int \cos (x) \sin ^2(x) \, dx}{a \sqrt {a \sec ^2(x)}} \\ & = \frac {\sec (x) \text {Subst}\left (\int x^2 \, dx,x,\sin (x)\right )}{a \sqrt {a \sec ^2(x)}} \\ & = \frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\tan ^3(x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(19)=38\).
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43
method | result | size |
derivativedivides | \(\frac {\tan \left (x \right )}{a \sqrt {a +a \tan \left (x \right )^{2}}}-a \left (\frac {\tan \left (x \right )}{3 a \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \tan \left (x \right )}{3 a^{2} \sqrt {a +a \tan \left (x \right )^{2}}}\right )\) | \(56\) |
default | \(\frac {\tan \left (x \right )}{a \sqrt {a +a \tan \left (x \right )^{2}}}-a \left (\frac {\tan \left (x \right )}{3 a \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \tan \left (x \right )}{3 a^{2} \sqrt {a +a \tan \left (x \right )^{2}}}\right )\) | \(56\) |
risch | \(\frac {i {\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {i {\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {i}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {i {\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}\) | \(149\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {a \tan \left (x\right )^{2} + a} \tan \left (x\right )^{3}}{3 \, {\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} \]
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\[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (x \right )}}{\left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {\sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )}{12 \, a^{\frac {3}{2}}} \]
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\tan \left (x\right )^{3}}{3 \, {\left (a \tan \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \]
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Time = 11.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {{\mathrm {tan}\left (x\right )}^3}{\left (3\,a\,{\mathrm {tan}\left (x\right )}^2+3\,a\right )\,\sqrt {a\,{\mathrm {tan}\left (x\right )}^2+a}} \]
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