Integrand size = 17, antiderivative size = 25 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {1}{\sqrt {a \sec ^2(x)}}+\frac {\sqrt {a \sec ^2(x)}}{a} \]
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Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3738, 4209, 45} \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {a \sec ^2(x)}}{a}+\frac {1}{\sqrt {a \sec ^2(x)}} \]
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Rule 45
Rule 3738
Rule 4209
Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan ^3(x)}{\sqrt {a \sec ^2(x)}} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {-1+x}{(a x)^{3/2}} \, dx,x,\sec ^2(x)\right ) \\ & = \frac {1}{2} a \text {Subst}\left (\int \left (-\frac {1}{(a x)^{3/2}}+\frac {1}{a \sqrt {a x}}\right ) \, dx,x,\sec ^2(x)\right ) \\ & = \frac {1}{\sqrt {a \sec ^2(x)}}+\frac {\sqrt {a \sec ^2(x)}}{a} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {(3+\cos (2 x)) \sqrt {a \sec ^2(x)}}{2 a} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\sqrt {a +a \tan \left (x \right )^{2}}}{a}+\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\) | \(26\) |
default | \(\frac {\sqrt {a +a \tan \left (x \right )^{2}}}{a}+\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\) | \(26\) |
risch | \(\frac {{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+1}{2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\tan \left (x\right )^{2} + 2}{\sqrt {a \tan \left (x\right )^{2} + a}} \]
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\[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\int \frac {\tan ^{3}{\left (x \right )}}{\sqrt {a \left (\tan ^{2}{\left (x \right )} + 1\right )}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {{\left (\sin \left (x\right )^{2} - 2\right )} \sqrt {\sin \left (x\right ) + 1} \sqrt {-\sin \left (x\right ) + 1}}{\sqrt {a} \sin \left (x\right )^{2} - \sqrt {a}} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {a \tan \left (x\right )^{2} + a} + \frac {a}{\sqrt {a \tan \left (x\right )^{2} + a}}}{a} \]
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Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {2}\,\left (\cos \left (2\,x\right )+3\right )}{2\,\sqrt {a}\,\sqrt {\cos \left (2\,x\right )+1}} \]
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