Integrand size = 16, antiderivative size = 24 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3738, 4207, 197} \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
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Rule 197
Rule 3738
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {a \sec ^2(c+d x)}} \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\tan (c+d x)}{d \sqrt {a \sec ^2(c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )}{d \sqrt {a +a \tan \left (d x +c \right )^{2}}}\) | \(25\) |
default | \(\frac {\tan \left (d x +c \right )}{d \sqrt {a +a \tan \left (d x +c \right )^{2}}}\) | \(25\) |
risch | \(-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{2 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}}+\frac {i}{2 \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(101\) |
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none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\sqrt {a \tan \left (d x + c\right )^{2} + a} \tan \left (d x + c\right )}{a d \tan \left (d x + c\right )^{2} + a d} \]
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\[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {a \tan ^{2}{\left (c + d x \right )} + a}}\, dx \]
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none
Time = 0.40 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\sin \left (d x + c\right )}{\sqrt {a} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.51 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=-\frac {2}{\sqrt {a} d {\left (\frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )} \]
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Time = 10.97 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx=\frac {\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )}{4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+3}}}{a\,d} \]
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