Integrand size = 21, antiderivative size = 492 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))} \] Output:
-(a^2-b^2)^(1/2)*(2*a^4-29*a^2*b^2+42*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c)) /(a^2-b^2)^(1/2))/a^8/d+1/8*b*(45*a^4-200*a^2*b^2+168*b^4)*arctanh(cos(d*x +c))/a^8/d-1/30*(91*a^4-645*a^2*b^2+630*b^4)*cot(d*x+c)/a^7/d+1/8*(8*a^4-7 9*a^2*b^2+84*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/b/d-1/30*(15*a^4-187*a^2*b^2+2 10*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^5/b^2/d-1/3*cot(d*x+c)*csc(d*x+c)/b/d/(a +b*sin(d*x+c))^2+1/12*a*cot(d*x+c)*csc(d*x+c)^2/b^2/d/(a+b*sin(d*x+c))^2+1 /60*(5*a^4-60*a^2*b^2+63*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^3/b^2/d/(a+b*sin(d *x+c))^2+7/20*b*cot(d*x+c)*csc(d*x+c)^3/a^2/d/(a+b*sin(d*x+c))^2-1/5*cot(d *x+c)*csc(d*x+c)^4/a/d/(a+b*sin(d*x+c))^2+1/12*(4*a^4-54*a^2*b^2+63*b^4)*c ot(d*x+c)*csc(d*x+c)^2/a^4/b^2/d/(a+b*sin(d*x+c))
Time = 2.02 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {3840 \left (2 a^6-31 a^4 b^2+71 a^2 b^4-42 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+480 b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^6(c+d x) \left (-784 a^6+3256 a^4 b^2+7860 a^2 b^4-12600 b^6+2 \left (384 a^6-2131 a^4 b^2-6315 a^2 b^4+9450 b^6\right ) \cos (2 (c+d x))+\left (-368 a^6+824 a^4 b^2+6060 a^2 b^4-7560 b^6\right ) \cos (4 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+1260 b^6 \cos (6 (c+d x))-8156 a^5 b \sin (c+d x)+42270 a^3 b^3 \sin (c+d x)-37800 a b^5 \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-20715 a^3 b^3 \sin (3 (c+d x))+18900 a b^5 \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))-3780 a b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{3840 a^8 d} \] Input:
Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]
Output:
((-3840*(2*a^6 - 31*a^4*b^2 + 71*a^2*b^4 - 42*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Cos[(c + d*x)/2]] - 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*L og[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(-784*a^6 + 3256*a ^4*b^2 + 7860*a^2*b^4 - 12600*b^6 + 2*(384*a^6 - 2131*a^4*b^2 - 6315*a^2*b ^4 + 9450*b^6)*Cos[2*(c + d*x)] + (-368*a^6 + 824*a^4*b^2 + 6060*a^2*b^4 - 7560*b^6)*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] + 4 2270*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*S in[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(3840*a^8*d)
Time = 4.48 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.23, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.190, Rules used = {3042, 3205, 27, 3042, 3534, 27, 3042, 3534, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^6 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3205 |
| \(\displaystyle \frac {\int \frac {4 \csc ^4(c+d x) \left (-5 \left (2 a^4-20 b^2 a^2+21 b^4\right ) \sin ^2(c+d x)-3 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)+3 \left (5 a^4-44 b^2 a^2+42 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{240 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) \left (-5 \left (2 a^4-20 b^2 a^2+21 b^4\right ) \sin ^2(c+d x)-3 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)+3 \left (5 a^4-44 b^2 a^2+42 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-5 \left (2 a^4-20 b^2 a^2+21 b^4\right ) \sin (c+d x)^2-3 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)+3 \left (5 a^4-44 b^2 a^2+42 b^4\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))^3}dx}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {2 \csc ^4(c+d x) \left (-4 \left (5 a^6-65 b^2 a^4+123 b^4 a^2-63 b^6\right ) \sin ^2(c+d x)-a b \left (20 a^4-41 b^2 a^2+21 b^4\right ) \sin (c+d x)+3 \left (10 a^6-114 b^2 a^4+209 b^4 a^2-105 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^4(c+d x) \left (-4 \left (5 a^6-65 b^2 a^4+123 b^4 a^2-63 b^6\right ) \sin ^2(c+d x)-a b \left (20 a^4-41 b^2 a^2+21 b^4\right ) \sin (c+d x)+3 \left (10 a^6-114 b^2 a^4+209 b^4 a^2-105 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-4 \left (5 a^6-65 b^2 a^4+123 b^4 a^2-63 b^6\right ) \sin (c+d x)^2-a b \left (20 a^4-41 b^2 a^2+21 b^4\right ) \sin (c+d x)+3 \left (10 a^6-114 b^2 a^4+209 b^4 a^2-105 b^6\right )}{\sin (c+d x)^4 (a+b \sin (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 \csc ^4(c+d x) \left (-5 \left (4 a^4-54 b^2 a^2+63 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+2 \left (15 a^4-187 b^2 a^2+210 b^4\right ) \left (a^2-b^2\right )^2-a b \left (10 a^2-21 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\csc ^4(c+d x) \left (-5 \left (4 a^4-54 b^2 a^2+63 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+2 \left (15 a^4-187 b^2 a^2+210 b^4\right ) \left (a^2-b^2\right )^2-a b \left (10 a^2-21 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {-5 \left (4 a^4-54 b^2 a^2+63 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+2 \left (15 a^4-187 b^2 a^2+210 b^4\right ) \left (a^2-b^2\right )^2-a b \left (10 a^2-21 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int -\frac {\csc ^3(c+d x) \left (-4 b \left (15 a^4-187 b^2 a^2+210 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+15 b \left (8 a^4-79 b^2 a^2+84 b^4\right ) \left (a^2-b^2\right )^2-a b^2 \left (62 a^2-105 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {\csc ^3(c+d x) \left (-4 b \left (15 a^4-187 b^2 a^2+210 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+15 b \left (8 a^4-79 b^2 a^2+84 b^4\right ) \left (a^2-b^2\right )^2-a b^2 \left (62 a^2-105 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {-4 b \left (15 a^4-187 b^2 a^2+210 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+15 b \left (8 a^4-79 b^2 a^2+84 b^4\right ) \left (a^2-b^2\right )^2-a b^2 \left (62 a^2-105 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-a \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^3-15 \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin ^2(c+d x) b^2+4 \left (a^2-b^2\right )^2 \left (91 a^4-645 b^2 a^2+630 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-a \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^3-15 \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin ^2(c+d x) b^2+4 \left (a^2-b^2\right )^2 \left (91 a^4-645 b^2 a^2+630 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {\int \frac {-a \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) b^3-15 \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin (c+d x)^2 b^2+4 \left (a^2-b^2\right )^2 \left (91 a^4-645 b^2 a^2+630 b^4\right ) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {\frac {\int -\frac {15 \csc (c+d x) \left (\left (a^2-b^2\right )^2 \left (45 a^4-200 b^2 a^2+168 b^4\right ) b^3+a \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b^2 \left (a^2-b^2\right )^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \int \frac {\csc (c+d x) \left (\left (a^2-b^2\right )^2 \left (45 a^4-200 b^2 a^2+168 b^4\right ) b^3+a \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \int \frac {\left (a^2-b^2\right )^2 \left (45 a^4-200 b^2 a^2+168 b^4\right ) b^3+a \left (a^2-b^2\right )^2 \left (8 a^4-79 b^2 a^2+84 b^4\right ) \sin (c+d x) b^2}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {4 b^2 \left (a^2-b^2\right )^3 \left (2 a^4-29 a^2 b^2+42 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {4 b^2 \left (a^2-b^2\right )^3 \left (2 a^4-29 a^2 b^2+42 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}+\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {8 b^2 \left (a^2-b^2\right )^3 \left (2 a^4-29 a^2 b^2+42 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^2 \left (a^2-b^2\right )^3 \left (2 a^4-29 a^2 b^2+42 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \int \csc (c+d x)dx}{a}+\frac {8 b^2 \left (a^2-b^2\right )^{5/2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {\frac {\frac {3 \left (-\frac {-\frac {-\frac {15 \left (\frac {8 b^2 \left (a^2-b^2\right )^{5/2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {4 b^2 \left (91 a^4-645 a^2 b^2+630 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x)}{a d}}{2 a}-\frac {15 b \left (8 a^4-79 a^2 b^2+84 b^4\right ) \left (a^2-b^2\right )^2 \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {2 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{a \left (a^2-b^2\right )}+\frac {5 \left (a^2-b^2\right ) \left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}}{a \left (a^2-b^2\right )}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))^2}}{60 a^2 b^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}\) |
Input:
Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]
Output:
-1/3*(Cot[c + d*x]*Csc[c + d*x])/(b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x ]*Csc[c + d*x]^3)/(20*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + (((5*a^4 - 60*a^2*b^2 + 63*b^4) *Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b*Sin[c + d*x])^2) + ((3*((-2*(a^2 - b^2)^2*(15*a^4 - 187*a^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(3 *a*d) - (-1/2*((-15*((8*b^2*(a^2 - b^2)^(5/2)*(2*a^4 - 29*a^2*b^2 + 42*b^4 )*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - (b^3*( a^2 - b^2)^2*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(a*d) ))/a - (4*b^2*(a^2 - b^2)^2*(91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c + d*x]) /(a*d))/a - (15*b*(a^2 - b^2)^2*(8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x] *Csc[c + d*x])/(2*a*d))/(3*a)))/(a*(a^2 - b^2)) + (5*(a^2 - b^2)*(4*a^4 - 54*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b*Sin[c + d*x] )))/(a*(a^2 - b^2)))/(60*a^2*b^2)
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 = 5.73 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{2}-\frac {7 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+8 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+24 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-40 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+22 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-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 ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{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 (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} 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 {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {3 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}}{2}-\frac {7 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+8 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+24 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-40 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+22 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-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 ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{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 (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} 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 | \(\text {Expression too large to display}\) | \(1254\) |
Input:
int(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/32/a^7*(1/5*tan(1/2*d*x+1/2*c)^5*a^4-3/2*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+8*a^2*b^2*tan(1/2*d*x+1/2*c)^3+24*a^3*b*tan( 1/2*d*x+1/2*c)^2-40*a*b^3*tan(1/2*d*x+1/2*c)^2+22*a^4*tan(1/2*d*x+1/2*c)-2 16*tan(1/2*d*x+1/2*c)*a^2*b^2+240*tan(1/2*d*x+1/2*c)*b^4)-1/160/a^3/tan(1/ 2*d*x+1/2*c)^5-1/96*(-7*a^2+24*b^2)/a^5/tan(1/2*d*x+1/2*c)^3-1/32*(22*a^4- 216*a^2*b^2+240*b^4)/a^7/tan(1/2*d*x+1/2*c)+3/64/a^4*b/tan(1/2*d*x+1/2*c)^ 4-1/4/a^6*b*(3*a^2-5*b^2)/tan(1/2*d*x+1/2*c)^2-1/8/a^8*b*(45*a^4-200*a^2*b ^2+168*b^4)*ln(tan(1/2*d*x+1/2*c))-2/a^8*(((5/2*a^5*b^2-19/2*a^3*b^4+7*a*b ^6)*tan(1/2*d*x+1/2*c)^3+1/2*b*(4*a^6-9*a^4*b^2-21*a^2*b^4+26*b^6)*tan(1/2 *d*x+1/2*c)^2+1/2*a*b^2*(11*a^4-49*a^2*b^2+38*b^4)*tan(1/2*d*x+1/2*c)+1/2* a^2*b*(4*a^4-17*a^2*b^2+13*b^4))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1 /2*c)+a)^2+1/2*(2*a^6-31*a^4*b^2+71*a^2*b^4-42*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. 1244 vs. \(2 (467) = 934\).
Time = 0.48 (sec) , antiderivative size = 2571, normalized size of antiderivative = 5.23 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
[-1/240*(8*(91*a^5*b^2 - 645*a^3*b^4 + 630*a*b^6)*cos(d*x + c)^7 - 4*(92*a ^7 + 67*a^5*b^2 - 3450*a^3*b^4 + 3780*a*b^6)*cos(d*x + c)^5 + 40*(14*a^7 - 37*a^5*b^2 - 303*a^3*b^4 + 378*a*b^6)*cos(d*x + c)^3 - 60*(2*(2*a^5*b - 2 9*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^6 - 4*a^5*b + 58*a^3*b^3 - 84*a*b^5 - 6 *(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^4 + 6*(2*a^5*b - 29*a^3*b^ 3 + 42*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 29*a^2*b^4 + 42*b^6)*cos(d*x + c)^6 - 2*a^6 + 27*a^4*b^2 - 13*a^2*b^4 - 42*b^6 - (2*a^6 - 23*a^4*b^2 - 45*a^2*b^4 + 126*b^6)*cos(d*x + c)^4 + (4*a^6 - 52*a^4*b^2 - 3*a^2*b^4 + 1 26*b^6)*cos(d*x + c)^2)*sin(d*x + c))*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)) - 60*(4*a^7 - 17*a^5*b^2 - 58*a^3*b^4 + 84*a*b^6)* cos(d*x + c) + 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 16 8*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 2 00*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^ 5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504* b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*(90*a^ 5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b...
\[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cot ^{6}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(cot(d*x+c)**6/(a+b*sin(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**6/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
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
Time = 0.27 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/960*(120*(45*a^4*b - 200*a^2*b^3 + 168*b^5)*log(abs(tan(1/2*d*x + 1/2*c )))/a^8 + 960*(2*a^6 - 31*a^4*b^2 + 71*a^2*b^4 - 42*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^8) + 960*(5*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 19*a ^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 14*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 4*a^6*b* tan(1/2*d*x + 1/2*c)^2 - 9*a^4*b^3*tan(1/2*d*x + 1/2*c)^2 - 21*a^2*b^5*tan (1/2*d*x + 1/2*c)^2 + 26*b^7*tan(1/2*d*x + 1/2*c)^2 + 11*a^5*b^2*tan(1/2*d *x + 1/2*c) - 49*a^3*b^4*tan(1/2*d*x + 1/2*c) + 38*a*b^6*tan(1/2*d*x + 1/2 *c) + 4*a^6*b - 17*a^4*b^3 + 13*a^2*b^5)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b* tan(1/2*d*x + 1/2*c) + a)^2*a^8) - (12330*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 5 4800*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 46032*b^5*tan(1/2*d*x + 1/2*c)^5 - 6 60*a^5*tan(1/2*d*x + 1/2*c)^4 + 6480*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 7200 *a*b^4*tan(1/2*d*x + 1/2*c)^4 - 720*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 1200*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 - 240*a^3*b^2 *tan(1/2*d*x + 1/2*c)^2 + 45*a^4*b*tan(1/2*d*x + 1/2*c) - 6*a^5)/(a^8*tan( 1/2*d*x + 1/2*c)^5) - (6*a^12*tan(1/2*d*x + 1/2*c)^5 - 45*a^11*b*tan(1/2*d *x + 1/2*c)^4 - 70*a^12*tan(1/2*d*x + 1/2*c)^3 + 240*a^10*b^2*tan(1/2*d*x + 1/2*c)^3 + 720*a^11*b*tan(1/2*d*x + 1/2*c)^2 - 1200*a^9*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*a^12*tan(1/2*d*x + 1/2*c) - 6480*a^10*b^2*tan(1/2*d*x + 1 /2*c) + 7200*a^8*b^4*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 22.16 (sec) , antiderivative size = 1614, normalized size of antiderivative = 3.28 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^6/(a + b*sin(c + d*x))^3,x)
Output:
tan(c/2 + (d*x)/2)^5/(160*a^3*d) - (tan(c/2 + (d*x)/2)^3*((a^2 + 4*b^2)/(3 2*a^5) + 1/(24*a^3) - (3*b^2)/(8*a^5)))/d + (tan(c/2 + (d*x)/2)*(1/(8*a^3) - (3*(a^2 + 4*b^2))/(32*a^5) - (6*b*((6*b*((3*(a^2 + 4*b^2))/(32*a^5) + 1 /(8*a^3) - (9*b^2)/(8*a^5)))/a - (384*a^2*b + 256*b^3)/(1024*a^6) + (9*b*( a^2 + 4*b^2))/(16*a^6)))/a + (3*(a^2 + 4*b^2)*((3*(a^2 + 4*b^2))/(32*a^5) + 1/(8*a^3) - (9*b^2)/(8*a^5)))/a^2 + (3*b*(384*a^2*b + 256*b^3))/(512*a^7 )))/d - (tan(c/2 + (d*x)/2)^3*((187*a^5*b)/15 - 14*a^3*b^3) + a^6/5 + tan( c/2 + (d*x)/2)^4*((263*a^6)/15 + 112*a^2*b^4 - (358*a^4*b^2)/3) + tan(c/2 + (d*x)/2)^5*(1216*a*b^5 + (1519*a^5*b)/6 - 1360*a^3*b^3) - tan(c/2 + (d*x )/2)^2*((29*a^6)/15 - (14*a^4*b^2)/5) + tan(c/2 + (d*x)/2)^8*(22*a^6 + 448 *b^6 - 368*a^2*b^4 - 56*a^4*b^2) + tan(c/2 + (d*x)/2)^6*((125*a^6)/3 + 217 6*b^6 - 2112*a^2*b^4 + 112*a^4*b^2) + (8*tan(c/2 + (d*x)/2)^7*(30*a^6*b + 104*b^7 + 36*a^2*b^5 - 149*a^4*b^3))/a - (7*a^5*b*tan(c/2 + (d*x)/2))/10)/ (d*(32*a^9*tan(c/2 + (d*x)/2)^5 + 32*a^9*tan(c/2 + (d*x)/2)^9 + tan(c/2 + (d*x)/2)^7*(64*a^9 + 128*a^7*b^2) + 128*a^8*b*tan(c/2 + (d*x)/2)^6 + 128*a ^8*b*tan(c/2 + (d*x)/2)^8)) + (tan(c/2 + (d*x)/2)^2*((3*b*((3*(a^2 + 4*b^2 ))/(32*a^5) + 1/(8*a^3) - (9*b^2)/(8*a^5)))/a - (384*a^2*b + 256*b^3)/(204 8*a^6) + (9*b*(a^2 + 4*b^2))/(32*a^6)))/d - (log(tan(c/2 + (d*x)/2))*(45*a ^4*b + 168*b^5 - 200*a^2*b^3))/(8*a^8*d) - (3*b*tan(c/2 + (d*x)/2)^4)/(64* a^4*d) - (atan((((-(a + b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (29*a^2*b^2)/...
Time = 0.24 (sec) , antiderivative size = 1241, normalized size of antiderivative = 2.52 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x)
Output:
( - 3840*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2) )*sin(c + d*x)**7*a**4*b**3 + 55680*sqrt(a**2 - b**2)*atan((tan((c + d*x)/ 2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**7*a**2*b**5 - 80640*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**7*b* *7 - 7680*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2 ))*sin(c + d*x)**6*a**5*b**2 + 111360*sqrt(a**2 - b**2)*atan((tan((c + d*x )/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**6*a**3*b**4 - 161280*sqrt(a** 2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**6 *a*b**6 - 3840*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**6*b + 55680*sqrt(a**2 - b**2)*atan((tan((c + d* x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**4*b**3 - 80640*sqrt(a** 2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5 *a**2*b**5 - 5824*cos(c + d*x)*sin(c + d*x)**6*a**5*b**3 + 41280*cos(c + d *x)*sin(c + d*x)**6*a**3*b**5 - 40320*cos(c + d*x)*sin(c + d*x)**6*a*b**7 - 9728*cos(c + d*x)*sin(c + d*x)**5*a**6*b**2 + 63600*cos(c + d*x)*sin(c + d*x)**5*a**4*b**4 - 60480*cos(c + d*x)*sin(c + d*x)**5*a**2*b**6 - 2944*c os(c + d*x)*sin(c + d*x)**4*a**7*b + 15328*cos(c + d*x)*sin(c + d*x)**4*a* *5*b**3 - 13440*cos(c + d*x)*sin(c + d*x)**4*a**3*b**5 - 3664*cos(c + d*x) *sin(c + d*x)**3*a**6*b**2 + 3360*cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 + 1408*cos(c + d*x)*sin(c + d*x)**2*a**7*b - 1344*cos(c + d*x)*sin(c + d...