Integrand size = 23, antiderivative size = 147 \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d} \]
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Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3965, 90, 52, 65, 213} \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{d} \]
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Rule 52
Rule 65
Rule 90
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-3 a^2 (a+a x)^{5/2}+\frac {a^2 (a+a x)^{5/2}}{x}+a (a+a x)^{7/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = -\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {a \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)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {2 \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d} \\ & = -\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.69 \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\frac {2 \sqrt {a (1+\sec (c+d x))} \left (-315 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )+\sqrt {1+\sec (c+d x)} \left (383-34 \sec (c+d x)-132 \sec ^2(c+d x)+5 \sec ^3(c+d x)+35 \sec ^4(c+d x)\right )\right )}{315 d \sqrt {1+\sec (c+d x)}} \]
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Time = 5.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {2 \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (315 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+383-34 \sec \left (d x +c \right )-132 \sec \left (d x +c \right )^{2}+5 \sec \left (d x +c \right )^{3}+35 \sec \left (d x +c \right )^{4}\right )}{315 d}\) | \(101\) |
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Time = 0.40 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.03 \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\left [\frac {315 \, \sqrt {a} \cos \left (d x + c\right )^{4} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{630 \, d \cos \left (d x + c\right )^{4}}, \frac {315 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{4} + 2 \, {\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}}\right ] \]
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\[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \tan ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99 \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\frac {315 \, \sqrt {a} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 630 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \frac {70 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{4}} - \frac {270 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a^{3}} + \frac {126 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}}}{a^{2}} + \frac {210 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}}}{a}}{315 \, d} \]
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\[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \tan \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^5\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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