Integrand size = 21, antiderivative size = 241 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^6 d}-\frac {\left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}+\frac {b \left (-9 a^2+4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (11 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]
-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^6/d+ 1/8*b*(15*a^4-20*a^2*b^2+8*b^4)*arctanh(cos(d*x+c))/a^6/d-1/15*(23*a^4-35* a^2*b^2+15*b^4)*cot(d*x+c)/a^5/d+1/8*b*(-9*a^2+4*b^2)*cot(d*x+c)*csc(d*x+c )/a^4/d+1/15*(11*a^2-5*b^2)*cot(d*x+c)*csc(d*x+c)^2/a^3/d+1/4*b*cot(d*x+c) *csc(d*x+c)^3/a^2/d-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d
Leaf count is larger than twice the leaf count of optimal. \(504\) vs. \(2(241)=482\).
Time = 1.01 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.09 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-1920 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-32 \left (23 a^5-35 a^3 b^2+15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )-270 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )+1800 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2400 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-1800 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2400 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+270 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-656 a^5 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+41 a^5 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-20 a^3 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 a^5 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+736 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-1120 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )+6 a^5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{960 a^6 d} \]
(-1920*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*Cot[(c + d*x)/2] - 270*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 + 1800*a^4*b*Log[Cos[(c + d*x)/2]] - 2400*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 1800*a^4*b*Log[Sin[(c + d*x)/2]] + 2400*a ^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 270*a^4*b*S ec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x )/2]^4 - 656*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d *x]^3*Sin[(c + d*x)/2]^4 + 41*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3 *b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d* x] + 736*a^5*Tan[(c + d*x)/2] - 1120*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4* Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)
Time = 2.16 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.43, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3205, 27, 3042, 3534, 25, 3042, 3534, 25, 3042, 3534, 27, 3042, 3480, 3042, 3139, 1083, 217, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^6 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3205 |
| \(\displaystyle \frac {\int \frac {2 \csc ^4(c+d x) \left (-5 \left (4 a^4-4 b^2 a^2+3 b^4\right ) \sin ^2(c+d x)-a b \left (10 a^2-b^2\right ) \sin (c+d x)+2 \left (15 a^4-22 b^2 a^2+10 b^4\right )\right )}{a+b \sin (c+d x)}dx}{40 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) \left (-5 \left (4 a^4-4 b^2 a^2+3 b^4\right ) \sin ^2(c+d x)-a b \left (10 a^2-b^2\right ) \sin (c+d x)+2 \left (15 a^4-22 b^2 a^2+10 b^4\right )\right )}{a+b \sin (c+d x)}dx}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-5 \left (4 a^4-4 b^2 a^2+3 b^4\right ) \sin (c+d x)^2-a b \left (10 a^2-b^2\right ) \sin (c+d x)+2 \left (15 a^4-22 b^2 a^2+10 b^4\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^3(c+d x) \left (a \left (28 a^2+5 b^2\right ) \sin (c+d x) b^2-4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b+15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^3(c+d x) \left (a \left (28 a^2+5 b^2\right ) \sin (c+d x) b^2-4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b+15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a \left (28 a^2+5 b^2\right ) \sin (c+d x) b^2-4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin (c+d x)^2 b+15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) b}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-a \left (41 a^2-20 b^2\right ) \sin (c+d x) b^3-15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin ^2(c+d x) b^2+8 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-a \left (41 a^2-20 b^2\right ) \sin (c+d x) b^3-15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin ^2(c+d x) b^2+8 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-a \left (41 a^2-20 b^2\right ) \sin (c+d x) b^3-15 \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin (c+d x)^2 b^2+8 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int -\frac {15 \csc (c+d x) \left (\left (15 a^4-20 b^2 a^2+8 b^4\right ) b^3+a \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \int \frac {\csc (c+d x) \left (\left (15 a^4-20 b^2 a^2+8 b^4\right ) b^3+a \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \int \frac {\left (15 a^4-20 b^2 a^2+8 b^4\right ) b^3+a \left (8 a^4-9 b^2 a^2+4 b^4\right ) \sin (c+d x) b^2}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \left (\frac {8 b^2 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \left (\frac {8 b^2 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \left (\frac {16 b^2 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \left (\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^2 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 \left (\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {16 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {-\frac {-\frac {-\frac {15 \left (\frac {16 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^3 \left (15 a^4-20 a^2 b^2+8 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {8 b^2 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-9 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{20 a^2 b^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\cot (c+d x) \csc (c+d x)}{b d}\) |
-((Cot[c + d*x]*Csc[c + d*x])/(b*d)) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(2* b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d) + ((-2*(15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - (-1/2*((-15*((16*b^2*(a^2 - b^2)^(5/2)*ArcTan[(2*b + 2 *a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - (b^3*(15*a^4 - 20*a^2*b ^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (8*b^2*(23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a*d))/a - (15*b*(8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[ c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a))/(20*a^2*b^2)
3.14.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^6, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(5*a*f*Sin[ e + f*x]^5)), x] + (Simp[Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*m* Sin[e + f*x]^2)), x] + Simp[a*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b ^2*f*m*(m - 1)*Sin[e + f*x]^3)), x] - Simp[b*(m - 4)*Cos[e + f*x]*((a + b*S in[e + f*x])^(m + 1)/(20*a^2*f*Sin[e + f*x]^4)), x] + Simp[1/(20*a^2*b^2*m* (m - 1)) Int[((a + b*Sin[e + f*x])^m/Sin[e + f*x]^4)*Simp[60*a^4 - 44*a^2 *b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f, m}, x] & & NeQ[a^2 - b^2, 0] && NeQ[m, 1] && IntegerQ[2*m]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.66 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.62
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+8 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-36 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (2 a^{2}-b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (15 a^{4}-20 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}+\frac {\left (-64 a^{6}+192 a^{4} b^{2}-192 a^{2} b^{4}+64 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{32 a^{6} \sqrt {a^{2}-b^{2}}}}{d}\) | \(390\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+8 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{5}}-\frac {1}{160 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+4 b^{2}}{96 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-36 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (2 a^{2}-b^{2}\right )}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (15 a^{4}-20 a^{2} b^{2}+8 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{6}}+\frac {\left (-64 a^{6}+192 a^{4} b^{2}-192 a^{2} b^{4}+64 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{32 a^{6} \sqrt {a^{2}-b^{2}}}}{d}\) | \(390\) |
| risch | \(\frac {-1200 i a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+560 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+135 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-60 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-120 i b^{4}+280 i a^{2} b^{2}+1600 i a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-150 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+120 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}-360 i a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-1120 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-1040 i a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-720 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+360 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+720 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+150 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}-120 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+480 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-184 i a^{4}-135 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+60 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a}{60 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a^{2} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{6} d}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}+\frac {2 \sqrt {-a^{2}+b^{2}}\, b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{4}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}-\frac {2 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{6}}+\frac {15 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a^{2} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{6} d}\) | \(806\) |
1/d*(1/32/a^5*(1/5*tan(1/2*d*x+1/2*c)^5*a^4-1/2*b*tan(1/2*d*x+1/2*c)^4*a^3 -7/3*a^4*tan(1/2*d*x+1/2*c)^3+4/3*a^2*b^2*tan(1/2*d*x+1/2*c)^3+8*a^3*b*tan (1/2*d*x+1/2*c)^2-4*a*b^3*tan(1/2*d*x+1/2*c)^2+22*tan(1/2*d*x+1/2*c)*a^4-3 6*tan(1/2*d*x+1/2*c)*a^2*b^2+16*tan(1/2*d*x+1/2*c)*b^4)-1/160/a/tan(1/2*d* x+1/2*c)^5-1/96*(-7*a^2+4*b^2)/a^3/tan(1/2*d*x+1/2*c)^3-1/32*(22*a^4-36*a^ 2*b^2+16*b^4)/a^5/tan(1/2*d*x+1/2*c)+1/64/a^2*b/tan(1/2*d*x+1/2*c)^4-1/8/a ^4*b*(2*a^2-b^2)/tan(1/2*d*x+1/2*c)^2-1/8/a^6*b*(15*a^4-20*a^2*b^2+8*b^4)* ln(tan(1/2*d*x+1/2*c))+1/32/a^6*(-64*a^6+192*a^4*b^2-192*a^2*b^4+64*b^6)/( a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (224) = 448\).
Time = 0.74 (sec) , antiderivative size = 1079, normalized size of antiderivative = 4.48 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
[-1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^3 - 120*((a^4 - 2*a^2*b^2 + b^4)*cos(d* x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^ 2)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2))*sin(d*x + c ) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*co s(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*c os(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15* a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8* b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*cos(d*x + c) ^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c )^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*sin(d*x + c)), -1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)* cos(d*x + c)^3 - 240*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2 *b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arc tan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) - 1 5*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos...
Timed out. \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (224) = 448\).
Time = 0.46 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.03 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1080 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {120 \, {\left (15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} - \frac {1920 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {4110 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5480 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1080 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
1/960*((6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*b*tan(1/2*d*x + 1/2*c)^4 - 7 0*a^4*tan(1/2*d*x + 1/2*c)^3 + 40*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 240*a^3 *b*tan(1/2*d*x + 1/2*c)^2 - 120*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^4*tan (1/2*d*x + 1/2*c) - 1080*a^2*b^2*tan(1/2*d*x + 1/2*c) + 480*b^4*tan(1/2*d* x + 1/2*c))/a^5 - 120*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^6 - 1920*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/2*(d *x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^6) + (4110*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 5480*a ^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 2192*b^5*tan(1/2*d*x + 1/2*c)^5 - 660*a^5* tan(1/2*d*x + 1/2*c)^4 + 1080*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 480*a*b^4*t an(1/2*d*x + 1/2*c)^4 - 240*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 120*a^2*b^3*tan (1/2*d*x + 1/2*c)^3 + 70*a^5*tan(1/2*d*x + 1/2*c)^2 - 40*a^3*b^2*tan(1/2*d *x + 1/2*c)^2 + 15*a^4*b*tan(1/2*d*x + 1/2*c) - 6*a^5)/(a^6*tan(1/2*d*x + 1/2*c)^5))/d
Time = 11.95 (sec) , antiderivative size = 1099, normalized size of antiderivative = 4.56 \[ \int \frac {\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
tan(c/2 + (d*x)/2)^5/(160*a*d) + (tan(c/2 + (d*x)/2)^2*(b/(32*a^2) + (b*(7 /(32*a) - b^2/(8*a^3)))/a))/d - (tan(c/2 + (d*x)/2)*(b^2/(8*a^3) - 11/(16* a) + (2*b*(b/(16*a^2) + (2*b*(7/(32*a) - b^2/(8*a^3)))/a))/a))/d - (tan(c/ 2 + (d*x)/2)^3*(7/(96*a) - b^2/(24*a^3)))/d - (b*tan(c/2 + (d*x)/2)^4)/(64 *a^2*d) - (log(tan(c/2 + (d*x)/2))*((15*a^4*b)/8 + b^5 - (5*a^2*b^3)/2))/( a^6*d) + (tan(c/2 + (d*x)/2)^2*((7*a^4)/3 - (4*a^2*b^2)/3) - a^4/5 - tan(c /2 + (d*x)/2)^4*(22*a^4 + 16*b^4 - 36*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(4*a *b^3 - 8*a^3*b) + (a^3*b*tan(c/2 + (d*x)/2))/2)/(32*a^5*d*tan(c/2 + (d*x)/ 2)^5) - (atan((((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 + 44*a^ 8*b^4 - 39*a^10*b^2)/(4*a^10) + ((2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2))/(4*a^9))*(-(a + b)^5*(a - b)^5)^(1/2))/a^6 + (tan(c/2 + (d* x)/2)*(31*a^10*b - 32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9))*1i)/a^6 + ((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16*a^6*b^6 + 44*a^8*b^4 - 39*a ^10*b^2)/(4*a^10) - ((2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^12 - 32*a^10*b^2 ))/(4*a^9))*(-(a + b)^5*(a - b)^5)^(1/2))/a^6 + (tan(c/2 + (d*x)/2)*(31*a^ 10*b - 32*a^4*b^7 + 96*a^6*b^5 - 98*a^8*b^3))/(4*a^9))*1i)/a^6)/((15*a^10* b - 8*b^11 + 44*a^2*b^9 - 99*a^4*b^7 + 113*a^6*b^5 - 65*a^8*b^3)/(2*a^10) + (tan(c/2 + (d*x)/2)*(16*a^10 - 8*b^10 + 42*a^2*b^8 - 94*a^4*b^6 + 110*a^ 6*b^4 - 66*a^8*b^2))/(2*a^9) - ((-(a + b)^5*(a - b)^5)^(1/2)*((8*a^12 - 16 *a^6*b^6 + 44*a^8*b^4 - 39*a^10*b^2)/(4*a^10) + ((2*a^2*b - (tan(c/2 + ...