Integrand size = 29, antiderivative size = 417 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}-\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d} \]
-2*b^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^ 8/d-1/16*b*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*arctanh(cos(d*x+c))/a^8/d+ 1/105*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(d*x+c)/a^7/d+1/16*b*(11 *a^4-18*a^2*b^2+8*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/d-1/105*(45*a^4-77*a^2*b^ 2+35*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^5/d-1/3*cot(d*x+c)*csc(d*x+c)^3/b/d+1/ 24*(8*a^4-13*a^2*b^2+6*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^4/b/d+1/4*a*cot(d*x+ c)*csc(d*x+c)^4/b^2/d-1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(d*x+c)*csc(d*x+ c)^4/a^3/b^2/d+1/6*b*cot(d*x+c)*csc(d*x+c)^5/a^2/d-1/7*cot(d*x+c)*csc(d*x+ c)^6/a/d
Time = 2.17 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-107520 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+3360 \left (-5 a^6 b+30 a^4 b^3-40 a^2 b^5+16 b^7\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3360 b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 a \cot (c+d x) \csc ^6(c+d x) \left (1200 a^6+8176 a^4 b^2-16240 a^2 b^4+8400 b^6+8 \left (225 a^6-1519 a^4 b^2+3115 a^2 b^4-1575 b^6\right ) \cos (2 (c+d x))+16 \left (45 a^6+329 a^4 b^2-665 a^2 b^4+315 b^6\right ) \cos (4 (c+d x))+120 a^6 \cos (6 (c+d x))-1288 a^4 b^2 \cos (6 (c+d x))+1960 a^2 b^4 \cos (6 (c+d x))-840 b^6 \cos (6 (c+d x))-5110 a^5 b \sin (c+d x)+13860 a^3 b^3 \sin (c+d x)-8400 a b^5 \sin (c+d x)+2135 a^5 b \sin (3 (c+d x))-7770 a^3 b^3 \sin (3 (c+d x))+4200 a b^5 \sin (3 (c+d x))-1155 a^5 b \sin (5 (c+d x))+1890 a^3 b^3 \sin (5 (c+d x))-840 a b^5 \sin (5 (c+d x))\right )}{53760 a^8 d} \]
(-107520*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 3360*(-5*a^6*b + 30*a^4*b^3 - 40*a^2*b^5 + 16*b^7)*Log[Cos[(c + d* x)/2]] + 3360*b*(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]^6*(1200*a^6 + 8176*a^4*b^2 - 16240* a^2*b^4 + 8400*b^6 + 8*(225*a^6 - 1519*a^4*b^2 + 3115*a^2*b^4 - 1575*b^6)* Cos[2*(c + d*x)] + 16*(45*a^6 + 329*a^4*b^2 - 665*a^2*b^4 + 315*b^6)*Cos[4 *(c + d*x)] + 120*a^6*Cos[6*(c + d*x)] - 1288*a^4*b^2*Cos[6*(c + d*x)] + 1 960*a^2*b^4*Cos[6*(c + d*x)] - 840*b^6*Cos[6*(c + d*x)] - 5110*a^5*b*Sin[c + d*x] + 13860*a^3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 2135*a^5* b*Sin[3*(c + d*x)] - 7770*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 1155*a^5*b*Sin[5*(c + d*x)] + 1890*a^3*b^3*Sin[5*(c + d*x)] - 840 *a*b^5*Sin[5*(c + d*x)]))/(53760*a^8*d)
Time = 3.37 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.13, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3375, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 25, 3042, 3534, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^8 (a+b \sin (c+d x))}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {6 \csc ^6(c+d x) \left (-14 \left (6 a^4-10 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)-a b \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 \left (35 a^4-60 b^2 a^2+28 b^4\right )\right )}{a+b \sin (c+d x)}dx}{504 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^6(c+d x) \left (-14 \left (6 a^4-10 b^2 a^2+5 b^4\right ) \sin ^2(c+d x)-a b \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 \left (35 a^4-60 b^2 a^2+28 b^4\right )\right )}{a+b \sin (c+d x)}dx}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-14 \left (6 a^4-10 b^2 a^2+5 b^4\right ) \sin (c+d x)^2-a b \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 \left (35 a^4-60 b^2 a^2+28 b^4\right )}{\sin (c+d x)^6 (a+b \sin (c+d x))}dx}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int -\frac {2 \csc ^5(c+d x) \left (a \left (10 a^2+7 b^2\right ) \sin (c+d x) b^2-6 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin ^2(c+d x) b+35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \int \frac {\csc ^5(c+d x) \left (a \left (10 a^2+7 b^2\right ) \sin (c+d x) b^2-6 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin ^2(c+d x) b+35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) b\right )}{a+b \sin (c+d x)}dx}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a \left (10 a^2+7 b^2\right ) \sin (c+d x) b^2-6 \left (35 a^4-60 b^2 a^2+28 b^4\right ) \sin (c+d x)^2 b+35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) b}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {2 \left (\frac {\int -\frac {3 \csc ^4(c+d x) \left (-a \left (25 a^2-14 b^2\right ) \sin (c+d x) b^3-35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin ^2(c+d x) b^2+8 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \int \frac {\csc ^4(c+d x) \left (-a \left (25 a^2-14 b^2\right ) \sin (c+d x) b^3-35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin ^2(c+d x) b^2+8 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^2\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \int \frac {-a \left (25 a^2-14 b^2\right ) \sin (c+d x) b^3-35 \left (8 a^4-13 b^2 a^2+6 b^4\right ) \sin (c+d x)^2 b^2+8 \left (45 a^4-77 b^2 a^2+35 b^4\right ) b^2}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (\frac {\int -\frac {\csc ^3(c+d x) \left (-16 \left (45 a^4-77 b^2 a^2+35 b^4\right ) \sin ^2(c+d x) b^3+105 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^3+a \left (120 a^4-133 b^2 a^2+70 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\int \frac {\csc ^3(c+d x) \left (-16 \left (45 a^4-77 b^2 a^2+35 b^4\right ) \sin ^2(c+d x) b^3+105 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^3+a \left (120 a^4-133 b^2 a^2+70 b^4\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\int \frac {-16 \left (45 a^4-77 b^2 a^2+35 b^4\right ) \sin (c+d x)^2 b^3+105 \left (11 a^4-18 b^2 a^2+8 b^4\right ) b^3+a \left (120 a^4-133 b^2 a^2+70 b^4\right ) \sin (c+d x) b^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {\int \frac {\csc ^2(c+d x) \left (105 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) b^4-a \left (285 a^4-574 b^2 a^2+280 b^4\right ) \sin (c+d x) b^3+16 \left (15 a^6-161 b^2 a^4+245 b^4 a^2-105 b^6\right ) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {\int \frac {105 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)^2 b^4-a \left (285 a^4-574 b^2 a^2+280 b^4\right ) \sin (c+d x) b^3+16 \left (15 a^6-161 b^2 a^4+245 b^4 a^2-105 b^6\right ) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {\frac {\int -\frac {105 \csc (c+d x) \left (b^3 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^4 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \int \frac {\csc (c+d x) \left (b^3 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^4 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \int \frac {b^3 \left (5 a^6-30 b^2 a^4+40 b^4 a^2-16 b^6\right )-a b^4 \left (11 a^4-18 b^2 a^2+8 b^4\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \left (\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^4 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \left (\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^4 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \left (\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^4 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \left (\frac {64 b^4 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {2 \left (-\frac {3 \left (-\frac {\frac {-\frac {105 \left (\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^4 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}\right )}{5 a}-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}}{84 a^2 b^2}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {-\frac {3 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {2 \left (-\frac {35 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \left (-\frac {8 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {\frac {-\frac {105 \left (-\frac {32 b^4 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {16 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {105 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}\right )}{4 a}\right )}{5 a}}{84 a^2 b^2}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}\) |
-1/3*(Cot[c + d*x]*Csc[c + d*x]^3)/(b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4) /(4*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d) - (Cot[c + d*x]*Csc [c + d*x]^6)/(7*a*d) + ((-3*(35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Cs c[c + d*x]^4)/(5*a*d) - (2*((-35*b*(8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d* x]*Csc[c + d*x]^3)/(4*a*d) - (3*((-8*b^2*(45*a^4 - 77*a^2*b^2 + 35*b^4)*Co t[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - (((-105*((-32*b^4*(a^2 - b^2)^(5/2)*A rcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - (b^3*(5*a ^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (16*b^2*(15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(a*d) )/(2*a) - (105*b^3*(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)))/(5*a))/(84*a^2*b^2)
3.14.30.3.1 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 = 0.86 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.45
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}}{7}-\frac {b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{3}-a^{6} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{4} b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 b \,a^{5} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{6} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {28 a^{4} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {16 a^{2} b^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 a^{5} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 a^{3} b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6}+88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{2}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{4}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{6}}{128 a^{7}}-\frac {2 b^{2} \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^{8} \sqrt {a^{2}-b^{2}}}-\frac {1}{896 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {-5 a^{2}+4 b^{2}}{640 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {9 a^{4}-28 a^{2} b^{2}+16 b^{4}}{384 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{6}+88 a^{4} b^{2}-144 a^{2} b^{4}+64 b^{6}}{128 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \left (3 a^{2}-2 b^{2}\right )}{128 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b \left (15 a^{4}-32 a^{2} b^{2}+16 b^{4}\right )}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{8}}}{d}\) | \(604\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}}{7}-\frac {b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{3}-a^{6} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{4} b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 b \,a^{5} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{6} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {28 a^{4} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {16 a^{2} b^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 a^{5} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 a^{3} b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6}+88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{2}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{4}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{6}}{128 a^{7}}-\frac {2 b^{2} \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^{8} \sqrt {a^{2}-b^{2}}}-\frac {1}{896 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {-5 a^{2}+4 b^{2}}{640 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {9 a^{4}-28 a^{2} b^{2}+16 b^{4}}{384 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{6}+88 a^{4} b^{2}-144 a^{2} b^{4}+64 b^{6}}{128 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \left (3 a^{2}-2 b^{2}\right )}{128 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b \left (15 a^{4}-32 a^{2} b^{2}+16 b^{4}\right )}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{8}}}{d}\) | \(604\) |
| risch | \(\text {Expression too large to display}\) | \(1188\) |
1/d*(1/128/a^7*(1/7*tan(1/2*d*x+1/2*c)^7*a^6-1/3*b*tan(1/2*d*x+1/2*c)^6*a^ 5-a^6*tan(1/2*d*x+1/2*c)^5+4/5*a^4*b^2*tan(1/2*d*x+1/2*c)^5+3*b*a^5*tan(1/ 2*d*x+1/2*c)^4-2*a^3*b^3*tan(1/2*d*x+1/2*c)^4+3*a^6*tan(1/2*d*x+1/2*c)^3-2 8/3*a^4*b^2*tan(1/2*d*x+1/2*c)^3+16/3*a^2*b^4*tan(1/2*d*x+1/2*c)^3-15*a^5* b*tan(1/2*d*x+1/2*c)^2+32*a^3*b^3*tan(1/2*d*x+1/2*c)^2-16*a*b^5*tan(1/2*d* x+1/2*c)^2-5*tan(1/2*d*x+1/2*c)*a^6+88*tan(1/2*d*x+1/2*c)*a^4*b^2-144*tan( 1/2*d*x+1/2*c)*a^2*b^4+64*tan(1/2*d*x+1/2*c)*b^6)-2*b^2*(a^6-3*a^4*b^2+3*a ^2*b^4-b^6)/a^8/(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))-1/896/a/tan(1/2*d*x+1/2*c)^7-1/640*(-5*a^2+4*b^2)/a^3/tan(1 /2*d*x+1/2*c)^5-1/384/a^5*(9*a^4-28*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^3-1 /128*(-5*a^6+88*a^4*b^2-144*a^2*b^4+64*b^6)/a^7/tan(1/2*d*x+1/2*c)+1/384/a ^2*b/tan(1/2*d*x+1/2*c)^6-1/128/a^4*b*(3*a^2-2*b^2)/tan(1/2*d*x+1/2*c)^4+1 /128/a^6*b*(15*a^4-32*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^2+1/16/a^8*b*(5*a ^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*ln(tan(1/2*d*x+1/2*c)))
Time = 0.94 (sec) , antiderivative size = 1645, normalized size of antiderivative = 3.94 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
[1/3360*(32*(15*a^7 - 161*a^5*b^2 + 245*a^3*b^4 - 105*a*b^6)*cos(d*x + c)^ 7 + 224*(58*a^5*b^2 - 100*a^3*b^4 + 45*a*b^6)*cos(d*x + c)^5 - 1120*(10*a^ 5*b^2 - 19*a^3*b^4 + 9*a*b^6)*cos(d*x + c)^3 + 1680*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*sqr t(-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) + 10 5*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7 - (5*a^6*b - 30*a^4*b^3 + 40 *a^2*b^5 - 16*b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*c os(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 105*(5*a^6*b - 3 0*a^4*b^3 + 40*a^2*b^5 - 16*b^7 - (5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16* b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d *x + c)^4 - 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^2) *log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3360*(a^5*b^2 - 2*a^3*b^4 + a *b^6)*cos(d*x + c) - 70*(3*(11*a^6*b - 18*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c )^5 - 8*(5*a^6*b - 12*a^4*b^3 + 6*a^2*b^5)*cos(d*x + c)^3 + 3*(5*a^6*b - 1 4*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^8*d*cos(d*x + c)^6 - 3*a^8*d*cos(d*x + c)^4 + 3*a^8*d*cos(d*x + c)^2 - a^8*d)*sin(d*x + c)...
Timed out. \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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 = 0.70 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
1/13440*((15*a^6*tan(1/2*d*x + 1/2*c)^7 - 35*a^5*b*tan(1/2*d*x + 1/2*c)^6 - 105*a^6*tan(1/2*d*x + 1/2*c)^5 + 84*a^4*b^2*tan(1/2*d*x + 1/2*c)^5 + 315 *a^5*b*tan(1/2*d*x + 1/2*c)^4 - 210*a^3*b^3*tan(1/2*d*x + 1/2*c)^4 + 315*a ^6*tan(1/2*d*x + 1/2*c)^3 - 980*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 + 560*a^2*b ^4*tan(1/2*d*x + 1/2*c)^3 - 1575*a^5*b*tan(1/2*d*x + 1/2*c)^2 + 3360*a^3*b ^3*tan(1/2*d*x + 1/2*c)^2 - 1680*a*b^5*tan(1/2*d*x + 1/2*c)^2 - 525*a^6*ta n(1/2*d*x + 1/2*c) + 9240*a^4*b^2*tan(1/2*d*x + 1/2*c) - 15120*a^2*b^4*tan (1/2*d*x + 1/2*c) + 6720*b^6*tan(1/2*d*x + 1/2*c))/a^7 + 840*(5*a^6*b - 30 *a^4*b^3 + 40*a^2*b^5 - 16*b^7)*log(abs(tan(1/2*d*x + 1/2*c)))/a^8 - 26880 *(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*(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) - (10890*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 65340*a^4*b^3*tan(1/2* d*x + 1/2*c)^7 + 87120*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 34848*b^7*tan(1/2* d*x + 1/2*c)^7 - 525*a^7*tan(1/2*d*x + 1/2*c)^6 + 9240*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 - 15120*a^3*b^4*tan(1/2*d*x + 1/2*c)^6 + 6720*a*b^6*tan(1/2*d* x + 1/2*c)^6 - 1575*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 3360*a^4*b^3*tan(1/2*d* x + 1/2*c)^5 - 1680*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 315*a^7*tan(1/2*d*x + 1/2*c)^4 - 980*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b^4*tan(1/2*d*x + 1/2*c)^4 + 315*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 210*a^4*b^3*tan(1/2*d*x + 1 /2*c)^3 - 105*a^7*tan(1/2*d*x + 1/2*c)^2 + 84*a^5*b^2*tan(1/2*d*x + 1/2...
Time = 12.46 (sec) , antiderivative size = 1513, normalized size of antiderivative = 3.63 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
(tan(c/2 + (d*x)/2)*(b^2/(32*a^3) - 5/(128*a) + (2*b*(b/(64*a^2) + (2*b*(5 /(128*a) - b^2/(32*a^3)))/a))/a + (2*b*(b/(64*a^2) - (2*b*(b^2/(32*a^3) - 9/(128*a) + (2*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/a + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/d + tan(c/2 + (d*x)/2)^7/(896*a *d) + (tan(c/2 + (d*x)/2)^4*(b/(256*a^2) + (b*(5/(128*a) - b^2/(32*a^3)))/ (2*a)))/d - (tan(c/2 + (d*x)/2)^2*(b/(128*a^2) - (b*(b^2/(32*a^3) - 9/(128 *a) + (2*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/a + (b*( 5/(128*a) - b^2/(32*a^3)))/a))/d - (tan(c/2 + (d*x)/2)^5*(1/(128*a) - b^2/ (160*a^3)))/d - (tan(c/2 + (d*x)/2)^3*(b^2/(96*a^3) - 3/(128*a) + (2*b*(b/ (64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/(3*a)))/d + (tan(c/2 + (d* x)/2)^2*(a^6 - (4*a^4*b^2)/5) - tan(c/2 + (d*x)/2)^3*(3*a^5*b - 2*a^3*b^3) - a^6/7 - tan(c/2 + (d*x)/2)^4*(3*a^6 + (16*a^2*b^4)/3 - (28*a^4*b^2)/3) + tan(c/2 + (d*x)/2)^5*(16*a*b^5 + 15*a^5*b - 32*a^3*b^3) + tan(c/2 + (d*x )/2)^6*(5*a^6 - 64*b^6 + 144*a^2*b^4 - 88*a^4*b^2) + (a^5*b*tan(c/2 + (d*x )/2))/3)/(128*a^7*d*tan(c/2 + (d*x)/2)^7) - (b*tan(c/2 + (d*x)/2)^6)/(384* a^2*d) + (log(tan(c/2 + (d*x)/2))*(5*a^6*b - 16*b^7 + 40*a^2*b^5 - 30*a^4* b^3))/(16*a^8*d) - (b^2*atan(((b^2*(-(a + b)^5*(a - b)^5)^(1/2)*((32*a^8*b ^8 - 88*a^10*b^6 + 78*a^12*b^4 - 21*a^14*b^2)/(8*a^14) + (tan(c/2 + (d*x)/ 2)*(5*a^14*b + 64*a^6*b^9 - 192*a^8*b^7 + 196*a^10*b^5 - 72*a^12*b^3))/(8* a^13) + (b^2*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8...