Integrand size = 15, antiderivative size = 12 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {1}{\sqrt {a \sec ^2(x)}} \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3738, 4209, 32} \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {1}{\sqrt {a \sec ^2(x)}} \]
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Rule 32
Rule 3738
Rule 4209
Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {1}{(a x)^{3/2}} \, dx,x,\sec ^2(x)\right ) \\ & = -\frac {1}{\sqrt {a \sec ^2(x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {1}{\sqrt {a \sec ^2(x)}} \]
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Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\) | \(13\) |
default | \(-\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\) | \(13\) |
risch | \(-\frac {{\mathrm e}^{2 i x}}{2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}-\frac {1}{2 \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}\) | \(65\) |
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {1}{\sqrt {a \tan \left (x\right )^{2} + a}} \]
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Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=- \frac {1}{\sqrt {a \tan ^{2}{\left (x \right )} + a}} \]
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\[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\int { \frac {\tan \left (x\right )}{\sqrt {a \tan \left (x\right )^{2} + a}} \,d x } \]
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {1}{\sqrt {a \tan \left (x\right )^{2} + a}} \]
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Time = 11.33 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {\tan (x)}{\sqrt {a+a \tan ^2(x)}} \, dx=-\frac {\sqrt {{\cos \left (x\right )}^2}}{\sqrt {a}} \]
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