Integrand size = 23, antiderivative size = 176 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d} \]
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Time = 0.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 491, 597, 536, 209} \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac {a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d} \]
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Rule 209
Rule 491
Rule 536
Rule 597
Rule 3972
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \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 )}{d} \\ & = -\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {\text {Subst}\left (\int \frac {-15 a-5 a^2 x^2}{x^4 \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 )}{5 d} \\ & = \frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {\text {Subst}\left (\int \frac {-105 a^2-45 a^3 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 )}{30 d} \\ & = -\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {\text {Subst}\left (\int \frac {-225 a^3-105 a^4 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 )}{60 d} \\ & = -\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {a^3 \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 )}{4 d}+\frac {\left (2 a^3\right ) \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 )}{d} \\ & = -\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d} \\ \end{align*}
Time = 6.40 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {\sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (\frac {1}{40} \cos ^3(c+d x) (160 \cos (c+d x)-7 (17+7 \cos (2 (c+d x)))) \csc ^5\left (\frac {1}{2} (c+d x)\right )+\frac {\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-8 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)}}{\sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}\right )}{32 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(147)=294\).
Time = 194.44 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (5 \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-80 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+80 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+98 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}-62 \cot \left (d x +c \right )^{3}-90 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )+70 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{40 d \left (\cos \left (d x +c \right )-1\right )}\) | \(372\) |
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Time = 0.40 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.69 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\left [\frac {5 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right ) + \sqrt {2} a^{2}\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 ) + 40 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\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 (49 \, a^{2} \cos \left (d x + c\right )^{3} - 80 \, a^{2} \cos \left (d x + c\right )^{2} + 35 \, a^{2} \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 (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}, -\frac {40 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\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 ) + 5 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right ) + \sqrt {2} a^{2}\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 ) + 2 \, {\left (49 \, a^{2} \cos \left (d x + c\right )^{3} - 80 \, a^{2} \cos \left (d x + c\right )^{2} + 35 \, a^{2} \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 (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}\right ] \]
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Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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