Integrand size = 27, antiderivative size = 363 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d} \]
2*b*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^7/d +1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*arctanh(cos(d*x+c))/a^7/d+1/15* b*(23*a^4-35*a^2*b^2+15*b^4)*cot(d*x+c)/a^6/d-1/16*(11*a^4-18*a^2*b^2+8*b^ 4)*cot(d*x+c)*csc(d*x+c)/a^5/d-1/2*cot(d*x+c)*csc(d*x+c)^2/b/d+1/30*(15*a^ 4-22*a^2*b^2+10*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^4/b/d+1/3*a*cot(d*x+c)*csc( d*x+c)^3/b^2/d-1/24*(8*a^4-13*a^2*b^2+6*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^3/b ^2/d+1/5*b*cot(d*x+c)*csc(d*x+c)^4/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a/d
Time = 1.73 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {7680 b \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+240 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 \left (-5 a^6+30 a^4 b^2-40 a^2 b^4+16 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a \cot (c+d x) \csc ^5(c+d x) \left (-295 a^5+570 a^3 b^2-360 a b^4+20 \left (7 a^5-42 a^3 b^2+24 a b^4\right ) \cos (2 (c+d x))-15 \left (11 a^5-18 a^3 b^2+8 a b^4\right ) \cos (4 (c+d x))+1168 a^4 b \sin (c+d x)-2320 a^2 b^3 \sin (c+d x)+1200 b^5 \sin (c+d x)-568 a^4 b \sin (3 (c+d x))+1240 a^2 b^3 \sin (3 (c+d x))-600 b^5 \sin (3 (c+d x))+184 a^4 b \sin (5 (c+d x))-280 a^2 b^3 \sin (5 (c+d x))+120 b^5 \sin (5 (c+d x))\right )}{3840 a^7 d} \]
(7680*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 240*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*Log[Cos[(c + d*x)/2]] + 240*(-5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6)*Log[Sin[(c + d*x)/2]] + 2* a*Cot[c + d*x]*Csc[c + d*x]^5*(-295*a^5 + 570*a^3*b^2 - 360*a*b^4 + 20*(7* a^5 - 42*a^3*b^2 + 24*a*b^4)*Cos[2*(c + d*x)] - 15*(11*a^5 - 18*a^3*b^2 + 8*a*b^4)*Cos[4*(c + d*x)] + 1168*a^4*b*Sin[c + d*x] - 2320*a^2*b^3*Sin[c + d*x] + 1200*b^5*Sin[c + d*x] - 568*a^4*b*Sin[3*(c + d*x)] + 1240*a^2*b^3* Sin[3*(c + d*x)] - 600*b^5*Sin[3*(c + d*x)] + 184*a^4*b*Sin[5*(c + d*x)] - 280*a^2*b^3*Sin[5*(c + d*x)] + 120*b^5*Sin[5*(c + d*x)]))/(3840*a^7*d)
Time = 2.78 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.13, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {3042, 3375, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^7 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {6 \csc ^5(c+d x) \left (-3 \left (10 a^4-15 b^2 a^2+8 b^4\right ) \sin ^2(c+d x)-a b \left (5 a^2-b^2\right ) \sin (c+d x)+5 \left (8 a^4-13 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{180 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^5(c+d x) \left (-3 \left (10 a^4-15 b^2 a^2+8 b^4\right ) \sin ^2(c+d x)-a b \left (5 a^2-b^2\right ) \sin (c+d x)+5 \left (8 a^4-13 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-3 \left (10 a^4-15 b^2 a^2+8 b^4\right ) \sin (c+d x)^2-a b \left (5 a^2-b^2\right ) \sin (c+d x)+5 \left (8 a^4-13 b^2 a^2+6 b^4\right )}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \csc ^4(c+d x) \left (a \left (5 a^2+2 b^2\right ) \sin (c+d x) b^2-5 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin ^2(c+d x) b+4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \int \frac {\csc ^4(c+d x) \left (a \left (5 a^2+2 b^2\right ) \sin (c+d x) b^2-5 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin ^2(c+d x) b+4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \int \frac {a \left (5 a^2+2 b^2\right ) \sin (c+d x) b^2-5 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin (c+d x)^2 b+4 \left (15 a^4-22 b^2 a^2+10 b^4\right ) b}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {3 \left (\frac {\int -\frac {\csc ^3(c+d x) \left (-a \left (19 a^2-10 b^2\right ) \sin (c+d x) b^3-8 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b^2+15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {\csc ^3(c+d x) \left (-a \left (19 a^2-10 b^2\right ) \sin (c+d x) b^3-8 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin ^2(c+d x) b^2+15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {-a \left (19 a^2-10 b^2\right ) \sin (c+d x) b^3-8 \left (15 a^4-22 b^2 a^2+10 b^4\right ) \sin (c+d x)^2 b^2+15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) b^3+16 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^3+a \left (75 a^4-82 b^2 a^2+40 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) b^3+16 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^3+a \left (75 a^4-82 b^2 a^2+40 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\int \frac {-15 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)^2 b^3+16 \left (23 a^4-35 b^2 a^2+15 b^4\right ) b^3+a \left (75 a^4-82 b^2 a^2+40 b^4\right ) \sin (c+d x) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {\int \frac {15 \csc (c+d x) \left (b^2 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^3 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {\csc (c+d x) \left (b^2 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^3 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {b^2 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^3 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^3 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {64 b^3 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{30 a^2 b^2}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {-\frac {5 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \left (-\frac {4 b \left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {-\frac {15 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\frac {15 \left (-\frac {32 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {16 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a d}}{2 a}}{3 a}\right )}{4 a}}{30 a^2 b^2}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}\) |
-1/2*(Cot[c + d*x]*Csc[c + d*x]^2)/(b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3) /(3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4)/(5*a^2*d) - (Cot[c + d*x]*Csc [c + d*x]^5)/(6*a*d) + ((-5*(8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[ c + d*x]^3)/(4*a*d) - (3*((-4*b*(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*((-32*b^3*(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^2*(5*a^6 - 3 0*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (16*b^ 3*(23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a*d))/a - (15*b^2*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a)))/(4*a))/( 30*a^2*b^2)
3.14.29.3.1 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[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] )^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d *Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) *(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin [e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) *(n + 2)*(m + n + 5)*(m + n + 6)) Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin [e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 )*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 0]
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 = 0.76 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.34
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {2 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b}{3}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{3}}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+8 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-44 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{6}}-\frac {1}{384 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+4 b^{2}}{256 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-32 a^{2} b^{2}+16 b^{4}}{128 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+120 a^{4} b^{2}-160 a^{2} b^{4}+64 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{7}}+\frac {b}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-4 b^{2}\right )}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-18 a^{2} b^{2}+8 b^{4}\right )}{16 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(486\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {2 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b}{3}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{3}}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{2}+8 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-44 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{6}}-\frac {1}{384 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+4 b^{2}}{256 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-32 a^{2} b^{2}+16 b^{4}}{128 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+120 a^{4} b^{2}-160 a^{2} b^{4}+64 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{7}}+\frac {b}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-4 b^{2}\right )}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-18 a^{2} b^{2}+8 b^{4}\right )}{16 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{7} \sqrt {a^{2}-b^{2}}}}{d}\) | \(486\) |
| risch | \(\text {Expression too large to display}\) | \(1065\) |
1/d*(1/64/a^6*(1/6*tan(1/2*d*x+1/2*c)^6*a^5-2/5*b*tan(1/2*d*x+1/2*c)^5*a^4 -3/2*tan(1/2*d*x+1/2*c)^4*a^5+tan(1/2*d*x+1/2*c)^4*a^3*b^2+14/3*tan(1/2*d* x+1/2*c)^3*a^4*b-8/3*tan(1/2*d*x+1/2*c)^3*a^2*b^3+15/2*tan(1/2*d*x+1/2*c)^ 2*a^5-16*tan(1/2*d*x+1/2*c)^2*a^3*b^2+8*a*b^4*tan(1/2*d*x+1/2*c)^2-44*a^4* b*tan(1/2*d*x+1/2*c)+72*tan(1/2*d*x+1/2*c)*a^2*b^3-32*tan(1/2*d*x+1/2*c)*b ^5)-1/384/a/tan(1/2*d*x+1/2*c)^6-1/256*(-6*a^2+4*b^2)/a^3/tan(1/2*d*x+1/2* c)^4-1/128/a^5*(15*a^4-32*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^2+1/64/a^7*(- 20*a^6+120*a^4*b^2-160*a^2*b^4+64*b^6)*ln(tan(1/2*d*x+1/2*c))+1/160/a^2*b/ tan(1/2*d*x+1/2*c)^5-1/96/a^4*b*(7*a^2-4*b^2)/tan(1/2*d*x+1/2*c)^3+1/16*b* (11*a^4-18*a^2*b^2+8*b^4)/a^6/tan(1/2*d*x+1/2*c)+2*b*(a^6-3*a^4*b^2+3*a^2* b^4-b^6)/a^7/(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. 689 vs. \(2 (340) = 680\).
Time = 0.98 (sec) , antiderivative size = 1462, normalized size of antiderivative = 4.03 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
[1/480*(30*(11*a^6 - 18*a^4*b^2 + 8*a^2*b^4)*cos(d*x + c)^5 - 80*(5*a^6 - 12*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c)^3 + 240*((a^4*b - 2*a^2*b^3 + b^5)*co s(d*x + c)^6 - a^4*b + 2*a^2*b^3 - b^5 - 3*(a^4*b - 2*a^2*b^3 + b^5)*cos(d *x + c)^4 + 3*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*l og(-((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)) + 30*(5*a^6 - 14*a^4*b^2 + 8*a^ 2*b^4)*cos(d*x + c) + 15*((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d *x + c)^6 - 5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6 - 3*(5*a^6 - 30*a^4*b ^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2 + 40*a^2* b^4 - 16*b^6)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 15*((5*a^6 - 3 0*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^6 - 5*a^6 + 30*a^4*b^2 - 40* a^2*b^4 + 16*b^6 - 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^4 + 3*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*cos(d*x + c)^2)*log(-1 /2*cos(d*x + c) + 1/2) - 32*((23*a^5*b - 35*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^5 - 5*(7*a^5*b - 13*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^3 + 15*(a^5*b - 2*a ^3*b^3 + a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*d*cos(d*x + c)^6 - 3*a^7* d*cos(d*x + c)^4 + 3*a^7*d*cos(d*x + c)^2 - a^7*d), 1/480*(30*(11*a^6 - 18 *a^4*b^2 + 8*a^2*b^4)*cos(d*x + c)^5 - 80*(5*a^6 - 12*a^4*b^2 + 6*a^2*b^4) *cos(d*x + c)^3 - 480*((a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^6 - a^4*b...
Timed out. \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.46 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.73 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2160 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {120 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} + \frac {3840 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\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^{7}} + \frac {1470 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8820 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11760 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4704 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2160 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 960 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{6}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
1/1920*((5*a^5*tan(1/2*d*x + 1/2*c)^6 - 12*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 45*a^5*tan(1/2*d*x + 1/2*c)^4 + 30*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 140*a^ 4*b*tan(1/2*d*x + 1/2*c)^3 - 80*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 225*a^5*t an(1/2*d*x + 1/2*c)^2 - 480*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*a*b^4*tan (1/2*d*x + 1/2*c)^2 - 1320*a^4*b*tan(1/2*d*x + 1/2*c) + 2160*a^2*b^3*tan(1 /2*d*x + 1/2*c) - 960*b^5*tan(1/2*d*x + 1/2*c))/a^6 - 120*(5*a^6 - 30*a^4* b^2 + 40*a^2*b^4 - 16*b^6)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 + 3840*(a^6* b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*(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 ^7) + (1470*a^6*tan(1/2*d*x + 1/2*c)^6 - 8820*a^4*b^2*tan(1/2*d*x + 1/2*c) ^6 + 11760*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 4704*b^6*tan(1/2*d*x + 1/2*c)^ 6 + 1320*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 2160*a^3*b^3*tan(1/2*d*x + 1/2*c)^ 5 + 960*a*b^5*tan(1/2*d*x + 1/2*c)^5 - 225*a^6*tan(1/2*d*x + 1/2*c)^4 + 48 0*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 - 240*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 14 0*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 80*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 45*a^ 6*tan(1/2*d*x + 1/2*c)^2 - 30*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 12*a^5*b*ta n(1/2*d*x + 1/2*c) - 5*a^6)/(a^7*tan(1/2*d*x + 1/2*c)^6))/d
Time = 12.59 (sec) , antiderivative size = 1289, normalized size of antiderivative = 3.55 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
tan(c/2 + (d*x)/2)^6/(384*a*d) + (tan(c/2 + (d*x)/2)^3*(b/(96*a^2) + (2*b* (3/(32*a) - b^2/(16*a^3)))/(3*a)))/d - (tan(c/2 + (d*x)/2)^4*(3/(128*a) - b^2/(64*a^3)))/d - (tan(c/2 + (d*x)/2)*(b/(32*a^2) - (2*b*(b^2/(16*a^3) - 15/(64*a) + (2*b*(b/(32*a^2) + (2*b*(3/(32*a) - b^2/(16*a^3)))/a))/a))/a + (2*b*(3/(32*a) - b^2/(16*a^3)))/a))/d - (tan(c/2 + (d*x)/2)^2*(b^2/(32*a^ 3) - 15/(128*a) + (b*(b/(32*a^2) + (2*b*(3/(32*a) - b^2/(16*a^3)))/a))/a)) /d - (tan(c/2 + (d*x)/2)^3*((14*a^4*b)/3 - (8*a^2*b^3)/3) + tan(c/2 + (d*x )/2)^4*(8*a*b^4 + (15*a^5)/2 - 16*a^3*b^2) - tan(c/2 + (d*x)/2)^5*(44*a^4* b + 32*b^5 - 72*a^2*b^3) + a^5/6 - tan(c/2 + (d*x)/2)^2*((3*a^5)/2 - a^3*b ^2) - (2*a^4*b*tan(c/2 + (d*x)/2))/5)/(64*a^6*d*tan(c/2 + (d*x)/2)^6) - (b *tan(c/2 + (d*x)/2)^5)/(160*a^2*d) - (log(tan(c/2 + (d*x)/2))*(5*a^6 - 16* b^6 + 40*a^2*b^4 - 30*a^4*b^2))/(16*a^7*d) + (b*atan(((b*(-(a + b)^5*(a - b)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^13 + 64*a^5*b^8 - 192*a^7*b^6 + 196* a^9*b^4 - 72*a^11*b^2))/(8*a^11) - (21*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*b^3)/(8*a^12) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^14 - 64*a^ 12*b^2))/(8*a^11))*(-(a + b)^5*(a - b)^5)^(1/2))/a^7)*1i)/a^7 - (b*(-(a + b)^5*(a - b)^5)^(1/2)*((21*a^13*b - 32*a^7*b^7 + 88*a^9*b^5 - 78*a^11*b^3) /(8*a^12) - (tan(c/2 + (d*x)/2)*(5*a^13 + 64*a^5*b^8 - 192*a^7*b^6 + 196*a ^9*b^4 - 72*a^11*b^2))/(8*a^11) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^ 14 - 64*a^12*b^2))/(8*a^11))*(-(a + b)^5*(a - b)^5)^(1/2))/a^7)*1i)/a^7...