Integrand size = 23, antiderivative size = 265 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {319 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {63 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 a^3 d}-\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{384 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d} \]
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Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483, 593, 597, 536, 209} \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac {319 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {63 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{128 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a \sec (c+d x)+a}}{48 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a \sec (c+d x)+a}}{192 a^3 d}-\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a \sec (c+d x)+a}}{384 a^3 d} \]
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Rule 209
Rule 483
Rule 536
Rule 593
Rule 597
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^3 d} \\ & = -\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {5 a-7 a^2 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{6 a^4 d} \\ & = -\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {a^2-95 a^3 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{48 a^5 d} \\ & = -\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{384 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d}-\frac {\text {Subst}\left (\int \frac {-189 a^3-573 a^4 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{192 a^6 d} \\ & = \frac {63 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 a^3 d}-\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{384 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d}+\frac {\text {Subst}\left (\int \frac {579 a^4-189 a^5 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{384 a^6 d} \\ & = \frac {63 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 a^3 d}-\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{384 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d}+\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^2 d}-\frac {319 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 a^2 d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {319 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {63 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 a^3 d}-\frac {191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{384 a^3 d}-\frac {19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{192 a^3 d}-\frac {\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {a+a \sec (c+d x)}}{48 a^3 d} \\ \end{align*}
Time = 5.56 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (-\left ((-58+487 \cos (c+d x)+698 \cos (2 (c+d x))+409 \cos (3 (c+d x))) \csc \left (\frac {1}{2} (c+d x)\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right )\right )-49152 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}+30624 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\sec (c+d x)} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{3072 d (a (1+\sec (c+d x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(588\) vs. \(2(229)=458\).
Time = 1.88 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.22
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 (-192 \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {11}{2}} \sin \left (d x +c \right )+192 \left (1-\cos \left (d x +c \right )\right )^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {9}{2}} \csc \left (d x +c \right )-216 \left (1-\cos \left (d x +c \right )\right )^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {7}{2}} \csc \left (d x +c \right )+24 \left (1-\cos \left (d x +c \right )\right )^{8} \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 )^{7}+252 \left (1-\cos \left (d x +c \right )\right )^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {5}{2}} \csc \left (d x +c \right )-164 \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 )^{5}-315 \left (1-\cos \left (d x +c \right )\right )^{2} \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}} \csc \left (d x +c \right )+563 \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 )^{3}-3072 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right ) \left (1-\cos \left (d x +c \right )\right )-1179 \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 )+3828 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right ) \left (1-\cos \left (d x +c \right )\right )\right )}{3072 d \,a^{3} \left (1-\cos \left (d x +c \right )\right )}\) | \(589\) |
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Time = 0.38 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.61 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [-\frac {957 \, \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 ) \sin \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 768 \, {\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 {8 \, a \cos \left (d x + c\right )^{3} - 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 )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 4 \, {\left (409 \, \cos \left (d x + c\right )^{4} + 349 \, \cos \left (d x + c\right )^{3} - 185 \, \cos \left (d x + c\right )^{2} - 189 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{1536 \, {\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 )} \sin \left (d x + c\right )}, -\frac {957 \, \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 {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 768 \, {\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 ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (409 \, \cos \left (d x + c\right )^{4} + 349 \, \cos \left (d x + c\right )^{3} - 185 \, \cos \left (d x + c\right )^{2} - 189 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{768 \, {\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 )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 1.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {31 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {291 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {96 \, \sqrt {2}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )} \sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{768 \, d} \]
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Timed out. \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^2}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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