Integrand size = 21, antiderivative size = 54 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2}{a d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3965, 53, 65, 213} \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2}{a d \sqrt {a \sec (c+d x)+a}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d} \]
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Rule 53
Rule 65
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2}{a d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {2}{a d \sqrt {a+a \sec (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a^2 d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2}{a d \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\sec (c+d x)\right )}{a d \sqrt {a (1+\sec (c+d x))}} \]
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Time = 0.87 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{a \sqrt {a +a \sec \left (d x +c \right )}}}{d}\) | \(45\) |
default | \(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {2}{a \sqrt {a +a \sec \left (d x +c \right )}}}{d}\) | \(45\) |
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (46) = 92\).
Time = 0.31 (sec) , antiderivative size = 244, normalized size of antiderivative = 4.52 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {\sqrt {a} {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {\sqrt {-a} {\left (\cos \left (d x + c\right ) + 1\right )} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) + 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a^{2} d \cos \left (d x + c\right ) + a^{2} d}\right ] \]
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\[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\tan {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {\log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a}}{d} \]
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Time = 0.93 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.61 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {2 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} + \frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{d} \]
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Time = 14.47 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2}{a\,d\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{a^{3/2}\,d} \]
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