Integrand size = 23, antiderivative size = 200 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 a^2 d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.22 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3965, 105, 157, 162, 65, 213} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {51}{32 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac {3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac {19}{48 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rule 65
Rule 105
Rule 157
Rule 162
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {a^4 \text {Subst}\left (\int \frac {1}{x (-a+a x)^2 (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac {a \text {Subst}\left (\int \frac {2 a^2+\frac {9 a^2 x}{2}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {-14 a^4-\frac {35 a^4 x}{4}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{14 a^2 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}-\frac {\text {Subst}\left (\int \frac {70 a^6-\frac {105 a^6 x}{8}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{70 a^5 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {-210 a^8+\frac {1995 a^8 x}{16}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{210 a^8 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {210 a^{10}-\frac {5355 a^{10} x}{32}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{210 a^{11} d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 {13 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{64 a d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 {13 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{32 a^2 d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.45 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {a \left (-14-13 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))+8 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))\right )}{28 d (-1+\sec (c+d x)) (a (1+\sec (c+d x)))^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(163)=326\).
Time = 1.76 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.64
method | result | size |
default | \(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (1365 \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 )+13440 \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 )+4095 \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}}+40320 \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}}+4095 \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}}+40320 \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}}+1365 \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 )-16034 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{2}+13440 \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}}-25280 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{2}+3164 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}+24080 \cot \left (d x +c \right )^{2}+10710 \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{6720 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\) | \(528\) |
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (161) = 322\).
Time = 0.38 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.74 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) + 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{13440 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}, -\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) - 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6720 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}\right ] \]
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\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {\frac {1365 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {13440 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} + \frac {105 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, \sqrt {2} {\left (15 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{36} - 84 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{37} - 385 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{38} - 2730 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{39}\right )}}{a^{42} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{6720 \, d} \]
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Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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