Integrand size = 17, antiderivative size = 59 \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=-\frac {1}{8} a \text {arctanh}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x) \]
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Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3738, 4210, 2691, 3853, 3855} \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=-\frac {1}{8} a \cos (x) \sqrt {a \sec ^2(x)} \text {arctanh}(\sin (x))+\frac {1}{4} a \tan (x) \sec ^2(x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \tan (x) \sqrt {a \sec ^2(x)} \]
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Rule 2691
Rule 3738
Rule 3853
Rule 3855
Rule 4210
Rubi steps \begin{align*} \text {integral}& = \int \left (a \sec ^2(x)\right )^{3/2} \tan ^2(x) \, dx \\ & = \left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec ^3(x) \tan ^2(x) \, dx \\ & = \frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x)-\frac {1}{4} \left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec ^3(x) \, dx \\ & = -\frac {1}{8} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x)-\frac {1}{8} \left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec (x) \, dx \\ & = -\frac {1}{8} a \text {arctanh}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.58 \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \left (a \sec ^2(x)\right )^{3/2} \left (-\text {arctanh}(\sin (x)) \cos ^3(x)-\cos (x) \sin (x)+2 \tan (x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\tan \left (x \right ) \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {a \tan \left (x \right ) \sqrt {a +a \tan \left (x \right )^{2}}}{8}-\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \left (x \right )+\sqrt {a +a \tan \left (x \right )^{2}}\right )}{8}\) | \(54\) |
default | \(\frac {\tan \left (x \right ) \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {a \tan \left (x \right ) \sqrt {a +a \tan \left (x \right )^{2}}}{8}-\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \left (x \right )+\sqrt {a +a \tan \left (x \right )^{2}}\right )}{8}\) | \(54\) |
risch | \(\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{6 i x}-7 \,{\mathrm e}^{4 i x}+7 \,{\mathrm e}^{2 i x}-1\right )}{4 \left ({\mathrm e}^{2 i x}+1\right )^{3}}-\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{4}+\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{4}\) | \(118\) |
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Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\frac {1}{16} \, a^{\frac {3}{2}} \log \left (2 \, a \tan \left (x\right )^{2} - 2 \, \sqrt {a \tan \left (x\right )^{2} + a} \sqrt {a} \tan \left (x\right ) + a\right ) + \frac {1}{8} \, {\left (2 \, a \tan \left (x\right )^{3} + a \tan \left (x\right )\right )} \sqrt {a \tan \left (x\right )^{2} + a} \]
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\[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\int \left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac {3}{2}} \tan ^{2}{\left (x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (47) = 94\).
Time = 0.52 (sec) , antiderivative size = 934, normalized size of antiderivative = 15.83 \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \, {\left (\sqrt {a \tan \left (x\right )^{2} + a} {\left (2 \, \tan \left (x\right )^{2} + 1\right )} \tan \left (x\right ) + \sqrt {a} \log \left ({\left | -\sqrt {a} \tan \left (x\right ) + \sqrt {a \tan \left (x\right )^{2} + a} \right |}\right )\right )} a \]
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Timed out. \[ \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx=\int {\mathrm {tan}\left (x\right )}^2\,{\left (a\,{\mathrm {tan}\left (x\right )}^2+a\right )}^{3/2} \,d x \]
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