Integrand size = 21, antiderivative size = 492 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))} \]
[Out]
Time = 1.48 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2805, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2} \]
[In]
[Out]
Rule 210
Rule 632
Rule 2739
Rule 2805
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (5 a^4-44 a^2 b^2+42 b^4\right )-12 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)-20 \left (2 a^4-20 a^2 b^2+21 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{240 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^4(c+d x) \left (24 \left (10 a^6-114 a^4 b^2+209 a^2 b^4-105 b^6\right )-8 a b \left (20 a^4-41 a^2 b^2+21 b^4\right ) \sin (c+d x)-32 \left (5 a^6-65 a^4 b^2+123 a^2 b^4-63 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{480 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (48 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right )-24 a b \left (10 a^2-21 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-120 \left (a^2-b^2\right )^2 \left (4 a^4-54 a^2 b^2+63 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{480 a^4 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-360 b \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right )+24 a b^2 \left (62 a^2-105 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+96 b \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^5 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (96 b^2 \left (a^2-b^2\right )^2 \left (91 a^4-645 a^2 b^2+630 b^4\right )-24 a b^3 \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-360 b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^6 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-360 b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right )-360 a b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^7 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^8}-\frac {\left (b \left (45 a^4-200 a^2 b^2+168 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^8} \\ & = \frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^8 d} \\ & = \frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^8 d} \\ & = -\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 1.12 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {3840 \left (2 a^6-31 a^4 b^2+71 a^2 b^4-42 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+480 b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^6(c+d x) \left (-784 a^6+3256 a^4 b^2+7860 a^2 b^4-12600 b^6+2 \left (384 a^6-2131 a^4 b^2-6315 a^2 b^4+9450 b^6\right ) \cos (2 (c+d x))+\left (-368 a^6+824 a^4 b^2+6060 a^2 b^4-7560 b^6\right ) \cos (4 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+1260 b^6 \cos (6 (c+d x))-8156 a^5 b \sin (c+d x)+42270 a^3 b^3 \sin (c+d x)-37800 a b^5 \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-20715 a^3 b^3 \sin (3 (c+d x))+18900 a b^5 \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))-3780 a b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{3840 a^8 d} \]
[In]
[Out]
Time = 1.28 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+8 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-216 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{7}}-\frac {1}{160 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+24 b^{2}}{96 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-216 a^{2} b^{2}+240 b^{4}}{32 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{64 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (3 a^{2}-5 b^{2}\right )}{4 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (45 a^{4}-200 a^{2} b^{2}+168 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{5} b^{2}-\frac {19}{2} a^{3} b^{4}+7 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{4}-49 a^{2} b^{2}+38 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{4}-17 a^{2} b^{2}+13 b^{4}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{6}-31 a^{4} b^{2}+71 a^{2} b^{4}-42 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8}}}{d}\) | \(559\) |
default | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+8 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-216 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+240 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{7}}-\frac {1}{160 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+24 b^{2}}{96 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-216 a^{2} b^{2}+240 b^{4}}{32 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{64 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (3 a^{2}-5 b^{2}\right )}{4 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (45 a^{4}-200 a^{2} b^{2}+168 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{5} b^{2}-\frac {19}{2} a^{3} b^{4}+7 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{4}-49 a^{2} b^{2}+38 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{4}-17 a^{2} b^{2}+13 b^{4}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{6}-31 a^{4} b^{2}+71 a^{2} b^{4}-42 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8}}}{d}\) | \(559\) |
risch | \(\text {Expression too large to display}\) | \(1251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (467) = 934\).
Time = 0.84 (sec) , antiderivative size = 2571, normalized size of antiderivative = 5.23 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.43 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Time = 12.44 (sec) , antiderivative size = 1614, normalized size of antiderivative = 3.28 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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