Integrand size = 23, antiderivative size = 262 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3965, 105, 156, 157, 162, 65, 213} \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac {761}{512 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac {7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac {83}{256 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rule 65
Rule 105
Rule 156
Rule 157
Rule 162
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {a^6 \text {Subst}\left (\int \frac {1}{x (-a+a x)^3 (a+a x)^{11/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {13 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{11/2}} \, dx,x,\sec (c+d x)\right )}{4 d} \\ & = -\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {231 a^4 x}{4}}{x (-a+a x) (a+a x)^{11/2}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}-\frac {\text {Subst}\left (\int \frac {-72 a^6-\frac {1791 a^6 x}{8}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{72 a^3 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {504 a^8+\frac {8505 a^8 x}{16}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{504 a^6 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {\text {Subst}\left (\int \frac {-2520 a^{10}-\frac {2205 a^{10} x}{32}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2520 a^9 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {7560 a^{12}-\frac {235305 a^{12} x}{64}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{7560 a^{12} d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {-7560 a^{14}+\frac {719145 a^{14} x}{128}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{7560 a^{15} d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 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^2 d}+\frac {263 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{1024 a d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 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^3 d}+\frac {263 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{512 a^2 d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 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 = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.38 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cot ^4(c+d x) \left (-450+263 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+378 \sec (c+d x)\right )}{288 d (a (1+\sec (c+d x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(563\) vs. \(2(217)=434\).
Time = 1.68 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.15
method | result | size |
default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (82845 \cos \left (d x +c \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+645120 \cos \left (d x +c \right )^{3} \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+248535 \cos \left (d x +c \right )^{2} \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+1935360 \cos \left (d x +c \right )^{2} \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}}+248535 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+1935360 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+82845 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+764402 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{4}+645120 \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}}+1183040 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{4}-807214 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{4}-2224080 \cot \left (d x +c \right )^{4}-378378 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )+1063440 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+479430 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{322560 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\) | \(564\) |
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Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (213) = 426\).
Time = 0.39 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.45 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 1.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {82845 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {645120 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {315 \, {\left (33 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 31 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} - \frac {8 \, \sqrt {2} {\left (35 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{56} - 225 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{57} + 1008 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{58} + 4410 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{59} + 31185 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{60}\right )}}{a^{63} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{322560 \, d} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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