Integrand size = 23, antiderivative size = 193 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d} \]
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Time = 0.18 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3965, 90, 52, 65, 213} \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 (a \sec (c+d x)+a)^{13/2}}{13 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{11/2}}{11 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}+\frac {2 a^2 \sqrt {a \sec (c+d x)+a}}{d}+\frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac {2 a (a \sec (c+d x)+a)^{3/2}}{3 d} \]
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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)^{9/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-3 a^2 (a+a x)^{9/2}+\frac {a^2 (a+a x)^{9/2}}{x}+a (a+a x)^{11/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = -\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{9/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a \text {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d} \\ & = -\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\frac {(a (1+\sec (c+d x)))^{5/2} \left (-2 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )+2 \sqrt {1+\sec (c+d x)}+\frac {2}{3} (1+\sec (c+d x))^{3/2}+\frac {2}{5} (1+\sec (c+d x))^{5/2}+\frac {2}{7} (1+\sec (c+d x))^{7/2}+\frac {2}{9} (1+\sec (c+d x))^{9/2}-\frac {6}{11} (1+\sec (c+d x))^{11/2}+\frac {2}{13} (1+\sec (c+d x))^{13/2}\right )}{d (1+\sec (c+d x))^{5/2}} \]
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Time = 262.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {2 a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (45045 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+71689+31723 \sec \left (d x +c \right )-12531 \sec \left (d x +c \right )^{2}-27095 \sec \left (d x +c \right )^{3}-4445 \sec \left (d x +c \right )^{4}+8505 \sec \left (d x +c \right )^{5}+3465 \sec \left (d x +c \right )^{6}\right )}{45045 d}\) | \(124\) |
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Time = 0.37 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.00 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\left [\frac {45045 \, a^{\frac {5}{2}} \cos \left (d x + c\right )^{6} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (71689 \, a^{2} \cos \left (d x + c\right )^{6} + 31723 \, a^{2} \cos \left (d x + c\right )^{5} - 12531 \, a^{2} \cos \left (d x + c\right )^{4} - 27095 \, a^{2} \cos \left (d x + c\right )^{3} - 4445 \, a^{2} \cos \left (d x + c\right )^{2} + 8505 \, a^{2} \cos \left (d x + c\right ) + 3465 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{90090 \, d \cos \left (d x + c\right )^{6}}, \frac {45045 \, \sqrt {-a} a^{2} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{6} + 2 \, {\left (71689 \, a^{2} \cos \left (d x + c\right )^{6} + 31723 \, a^{2} \cos \left (d x + c\right )^{5} - 12531 \, a^{2} \cos \left (d x + c\right )^{4} - 27095 \, a^{2} \cos \left (d x + c\right )^{3} - 4445 \, a^{2} \cos \left (d x + c\right )^{2} + 8505 \, a^{2} \cos \left (d x + c\right ) + 3465 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{45045 \, d \cos \left (d x + c\right )^{6}}\right ] \]
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Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94 \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\frac {45045 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 18018 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}} + \frac {6930 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {13}{2}}}{a^{4}} - \frac {24570 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {11}{2}}}{a^{3}} + \frac {10010 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{2}} + \frac {12870 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a} + 30030 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a + 90090 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a^{2}}{45045 \, d} \]
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\[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \tan \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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