Integrand size = 23, antiderivative size = 280 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {151 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {105 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 d}-\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d} \]
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Time = 0.32 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483, 593, 597, 536, 209} \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {151 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{128 \sqrt {2} d}-\frac {23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac {105 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{128 d} \]
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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^6 \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 )}{a^2 d} \\ & = -\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}-\frac {\text {Subst}\left (\int \frac {-a-9 a^2 x^2}{x^6 \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 )}{4 a^3 d} \\ & = -\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}-\frac {\text {Subst}\left (\int \frac {-87 a^2-119 a^3 x^2}{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 )}{16 a^4 d} \\ & = \frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}+\frac {\text {Subst}\left (\int \frac {-115 a^3-435 a^4 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 )}{160 a^4 d} \\ & = -\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}-\frac {\text {Subst}\left (\int \frac {1575 a^4-345 a^5 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 )}{960 a^4 d} \\ & = -\frac {105 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 d}-\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}+\frac {\text {Subst}\left (\int \frac {5415 a^5+1575 a^6 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 )}{1920 a^4 d} \\ & = -\frac {105 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 d}-\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d}-\frac {(151 a) \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 d}+\frac {(2 a) \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 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {151 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {105 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{128 d}-\frac {23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac {87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac {17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{32 a^2 d}-\frac {\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{16 a^2 d} \\ \end{align*}
Time = 4.66 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.75 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\sqrt {a (1+\sec (c+d x))} \left (-\frac {(5207+172 \cos (c+d x)-4572 \cos (2 (c+d x))-556 \cos (3 (c+d x))+2821 \cos (4 (c+d x))) \csc ^5(c+d x)}{4 \sqrt {\sec (c+d x)}}-7680 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}+2265 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{3840 d \sqrt {\sec (c+d x)}} \]
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Time = 2.65 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\left (2265 \sin \left (d x +c \right )^{5} \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-7680 \sin \left (d x +c \right )^{5} \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-5642 \cos \left (d x +c \right )^{5}+556 \cos \left (d x +c \right )^{4}+7928 \cos \left (d x +c \right )^{3}-460 \cos \left (d x +c \right )^{2}-3150 \cos \left (d x +c \right )\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \csc \left (d x +c \right )^{5}}{3840 d}\) | \(226\) |
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Time = 0.42 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.31 \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {2265 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {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 ) + 3840 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 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 (2821 \, \cos \left (d x + c\right )^{5} - 278 \, \cos \left (d x + c\right )^{4} - 3964 \, \cos \left (d x + c\right )^{3} + 230 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{7680 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )}, -\frac {3840 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 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 ) + 2265 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {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 (2821 \, \cos \left (d x + c\right )^{5} - 278 \, \cos \left (d x + c\right )^{4} - 3964 \, \cos \left (d x + c\right )^{3} + 230 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3840 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \cot ^{6}{\left (c + d x \right )}\, dx \]
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Timed out. \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int \cot ^6(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^6\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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