Integrand size = 23, antiderivative size = 78 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3965, 81, 52, 65, 213} \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {a \sec (c+d x)+a}}{a d} \]
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-a+a x) \sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = -\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 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 d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \left (-2-\sec (c+d x)+\sec ^2(c+d x)+3 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) \sqrt {1+\sec (c+d x)}\right )}{3 d \sqrt {a (1+\sec (c+d x))}} \]
[In]
[Out]
Time = 3.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (3 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+2-\sec \left (d x +c \right )\right )}{3 d a}\) | \(74\) |
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.09 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\left [\frac {3 \, \sqrt {a} \cos \left (d x + c\right ) \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (2 \, \cos \left (d x + c\right ) - 1\right )}}{6 \, a d \cos \left (d x + c\right )}, -\frac {3 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right ) + 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (2 \, \cos \left (d x + c\right ) - 1\right )}}{3 \, a d \cos \left (d x + c\right )}\right ] \]
[In]
[Out]
\[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\tan ^{3}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {\frac {3 \, \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}}}{a^{2}} + \frac {6 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}}}{a}}{3 \, d} \]
[In]
[Out]
none
Time = 1.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.62 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (\frac {3 \, \sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a + 2 \, a^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )}}{3 \, a d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
[In]
[Out]