Integrand size = 21, antiderivative size = 144 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 a^2 d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3965, 87, 157, 162, 65, 213} \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {7}{4 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {1}{2 a d (a \sec (c+d x)+a)^{3/2}}-\frac {1}{5 d (a \sec (c+d x)+a)^{5/2}} \]
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Rule 65
Rule 87
Rule 157
Rule 162
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \text {Subst}\left (\int \frac {1}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {\text {Subst}\left (\int \frac {2 a^2-a^2 x}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2 a d} \\ & = -\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {-6 a^4+\frac {9 a^4 x}{2}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{6 a^4 d} \\ & = -\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {6 a^6-\frac {21 a^6 x}{4}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{6 a^7 d} \\ & = -\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 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 {\text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{8 a d} \\ & = -\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 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 {\text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{4 a^2 d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{5 d (a+a \sec (c+d x))^{5/2}}-\frac {1}{2 a d (a+a \sec (c+d x))^{3/2}}-\frac {7}{4 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 = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1+\sec (c+d x))\right )-2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1+\sec (c+d x)\right )}{5 d (a (1+\sec (c+d x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(377\) vs. \(2(115)=230\).
Time = 1.73 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.62
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \left (-15 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}}+15 \left (1-\cos \left (d x +c \right )\right )^{6} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{6}+21 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}}-87 \left (1-\cos \left (d x +c \right )\right )^{4} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{4}-35 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}+269 \left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{2}+840 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}{2}\right )+105 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right )-932 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right )}{840 d \,a^{3}}\) | \(378\) |
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (115) = 230\).
Time = 0.36 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.98 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [\frac {5 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \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 ) + 40 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \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 (49 \, \cos \left (d x + c\right )^{3} + 80 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{80 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, 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 {5 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \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 ) - 40 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \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 (49 \, \cos \left (d x + c\right )^{3} + 80 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{40 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, 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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\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.93 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.43 \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {5 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {80 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {\sqrt {2} {\left ({\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{20} + 5 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{21} + 35 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{22}\right )}}{a^{25} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{40 \, d} \]
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Timed out. \[ \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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