Integrand size = 23, antiderivative size = 238 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3965, 105, 156, 157, 162, 65, 213} \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac {15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac {53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac {309}{256 a d \sqrt {a \sec (c+d x)+a}} \]
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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)^{9/2}} \, 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))^{7/2}}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {11 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{9/2}} \, 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))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {171 a^4 x}{4}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac {\text {Subst}\left (\int \frac {-56 a^6-\frac {973 a^6 x}{8}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{56 a^3 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}+\frac {\text {Subst}\left (\int \frac {280 a^8+\frac {2625 a^8 x}{16}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{280 a^6 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {-840 a^{10}+\frac {5565 a^{10} x}{32}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{840 a^9 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {840 a^{12}-\frac {32445 a^{12} x}{64}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{840 a^{12} d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {203 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{512 d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a 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^2 d}+\frac {203 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{256 a d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a 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 = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.42 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\cot ^4(c+d x) \left (-322+203 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+266 \sec (c+d x)\right )}{224 d (a (1+\sec (c+d x)))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(197)=394\).
Time = 1.82 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.83
method | result | size |
default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (4263 \cos \left (d x +c \right )^{2} \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+8526 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+21504 \cos \left (d x +c \right )^{2} \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+4263 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+43008 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+21504 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+20726 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{4}+16074 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{4}-33076 \cot \left (d x +c \right )^{4}-28476 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )+14462 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+12978 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{10752 d \,a^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\) | \(436\) |
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Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (193) = 386\).
Time = 0.40 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.52 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {2 \, \sqrt {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 ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) + 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \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 \, {\left (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{21504 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {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 )}{a \cos \left (d x + c\right ) + a}\right ) - 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \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 \, {\left (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{10752 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]
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\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{5}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 1.01 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {4263 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21504 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21 \, {\left (29 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{30} - 21 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{31} - 112 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{32} - 882 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{33}\right )}}{a^{35} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{10752 \, d} \]
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Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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