Integrand size = 23, antiderivative size = 54 \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d} \]
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Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3965, 81, 65, 213} \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{a^2 d} \]
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Rule 65
Rule 81
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-a+a x}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {2 \sqrt {a+a \sec (c+d x)}}{a^2 d}-\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 \sqrt {a+a \sec (c+d x)}}{a^2 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \left (1+\sec (c+d x)+\text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) \sqrt {1+\sec (c+d x)}\right )}{a d \sqrt {a (1+\sec (c+d x))}} \]
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Time = 3.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-1\right )}{d \,a^{2}}\) | \(65\) |
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Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.54 \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {\sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{2 \, a^{2} d}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{a^{2} d}\right ] \]
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\[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.31 \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {\frac {\log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}}}{a^{2}}}{d} \]
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Time = 1.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.57 \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\sqrt {2}}{\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )}}{a d} \]
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Timed out. \[ \int \frac {\tan ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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