Integrand size = 27, antiderivative size = 480 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))} \] Output:
2*b*(2*a^2-7*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2 )^(1/2))/a^8/d+1/16*(5*a^6-90*a^4*b^2+200*a^2*b^4-112*b^6)*arctanh(cos(d*x +c))/a^8/d+1/15*b*(61*a^4-170*a^2*b^2+105*b^4)*cot(d*x+c)/a^7/d-1/16*(27*a ^4-86*a^2*b^2+56*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/d+1/15*(15*a^4-52*a^2*b^2+ 35*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^5/b/d-1/24*(16*a^4-61*a^2*b^2+42*b^4)*co t(d*x+c)*csc(d*x+c)^3/a^4/b^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2/b/d/(a+b*sin(d *x+c))+1/6*a*cot(d*x+c)*csc(d*x+c)^3/b^2/d/(a+b*sin(d*x+c))+1/10*(5*a^4-20 *a^2*b^2+14*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^3/b^2/d/(a+b*sin(d*x+c))+7/30*b *cot(d*x+c)*csc(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))-1/6*cot(d*x+c)*csc(d*x+c)^ 5/a/d/(a+b*sin(d*x+c))
Time = 3.54 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {15360 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+480 \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+480 \left (-5 a^6+90 a^4 b^2-200 a^2 b^4+112 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 a \cot (c+d x) \csc ^6(c+d x) \left (590 a^6-6956 a^4 b^2+15280 a^2 b^4-8400 b^6-8 \left (35 a^6-1289 a^4 b^2+2830 a^2 b^4-1575 b^6\right ) \cos (2 (c+d x))+\left (330 a^6-3844 a^4 b^2+8720 a^2 b^4-5040 b^6\right ) \cos (4 (c+d x))+488 a^4 b^2 \cos (6 (c+d x))-1360 a^2 b^4 \cos (6 (c+d x))+840 b^6 \cos (6 (c+d x))-3942 a^5 b \sin (c+d x)+12620 a^3 b^3 \sin (c+d x)-8400 a b^5 \sin (c+d x)+1967 a^5 b \sin (3 (c+d x))-6590 a^3 b^3 \sin (3 (c+d x))+4200 a b^5 \sin (3 (c+d x))-571 a^5 b \sin (5 (c+d x))+1430 a^3 b^3 \sin (5 (c+d x))-840 a b^5 \sin (5 (c+d x))\right )}{b+a \csc (c+d x)}}{7680 a^8 d} \] Input:
Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
(15360*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2]) /Sqrt[a^2 - b^2]] + 480*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*Log[C os[(c + d*x)/2]] + 480*(-5*a^6 + 90*a^4*b^2 - 200*a^2*b^4 + 112*b^6)*Log[S in[(c + d*x)/2]] - (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(590*a^6 - 6956*a^4*b^ 2 + 15280*a^2*b^4 - 8400*b^6 - 8*(35*a^6 - 1289*a^4*b^2 + 2830*a^2*b^4 - 1 575*b^6)*Cos[2*(c + d*x)] + (330*a^6 - 3844*a^4*b^2 + 8720*a^2*b^4 - 5040* b^6)*Cos[4*(c + d*x)] + 488*a^4*b^2*Cos[6*(c + d*x)] - 1360*a^2*b^4*Cos[6* (c + d*x)] + 840*b^6*Cos[6*(c + d*x)] - 3942*a^5*b*Sin[c + d*x] + 12620*a^ 3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 1967*a^5*b*Sin[3*(c + d*x)] - 6590*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 571*a^5*b *Sin[5*(c + d*x)] + 1430*a^3*b^3*Sin[5*(c + d*x)] - 840*a*b^5*Sin[5*(c + d *x)]))/(b + a*Csc[c + d*x]))/(7680*a^8*d)
Time = 4.18 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.21, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3375, 27, 3042, 3534, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^7 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {12 \csc ^5(c+d x) \left (20 a^4-65 b^2 a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a+42 b^4-5 \left (3 a^4-10 b^2 a^2+7 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{360 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc ^5(c+d x) \left (20 a^4-65 b^2 a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a+42 b^4-5 \left (3 a^4-10 b^2 a^2+7 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {20 a^4-65 b^2 a^2-b \left (5 a^2-2 b^2\right ) \sin (c+d x) a+42 b^4-5 \left (3 a^4-10 b^2 a^2+7 b^4\right ) \sin (c+d x)^2}{\sin (c+d x)^5 (a+b \sin (c+d x))^2}dx}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^5(c+d x) \left (-12 \left (5 a^6-25 b^2 a^4+34 b^4 a^2-14 b^6\right ) \sin ^2(c+d x)-a b \left (10 a^4-17 b^2 a^2+7 b^4\right ) \sin (c+d x)+5 \left (16 a^6-77 b^2 a^4+103 b^4 a^2-42 b^6\right )\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-12 \left (5 a^6-25 b^2 a^4+34 b^4 a^2-14 b^6\right ) \sin (c+d x)^2-a b \left (10 a^4-17 b^2 a^2+7 b^4\right ) \sin (c+d x)+5 \left (16 a^6-77 b^2 a^4+103 b^4 a^2-42 b^6\right )}{\sin (c+d x)^5 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {3 \csc ^4(c+d x) \left (-a \left (15 a^4-29 b^2 a^2+14 b^4\right ) \sin (c+d x) b^2-5 \left (16 a^6-77 b^2 a^4+103 b^4 a^2-42 b^6\right ) \sin ^2(c+d x) b+8 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \frac {\csc ^4(c+d x) \left (-a \left (15 a^4-29 b^2 a^2+14 b^4\right ) \sin (c+d x) b^2-5 \left (16 a^6-77 b^2 a^4+103 b^4 a^2-42 b^6\right ) \sin ^2(c+d x) b+8 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) b\right )}{a+b \sin (c+d x)}dx}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \frac {-a \left (15 a^4-29 b^2 a^2+14 b^4\right ) \sin (c+d x) b^2-5 \left (16 a^6-77 b^2 a^4+103 b^4 a^2-42 b^6\right ) \sin (c+d x)^2 b+8 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) b}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {\int -\frac {\csc ^3(c+d x) \left (-a \left (83 a^4-153 b^2 a^2+70 b^4\right ) \sin (c+d x) b^3-16 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) \sin ^2(c+d x) b^2+15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\int \frac {\csc ^3(c+d x) \left (-a \left (83 a^4-153 b^2 a^2+70 b^4\right ) \sin (c+d x) b^3-16 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) \sin ^2(c+d x) b^2+15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) b^2\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\int \frac {-a \left (83 a^4-153 b^2 a^2+70 b^4\right ) \sin (c+d x) b^3-16 \left (15 a^6-67 b^2 a^4+87 b^4 a^2-35 b^6\right ) \sin (c+d x)^2 b^2+15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) b^2}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin ^2(c+d x) b^3+16 \left (61 a^6-231 b^2 a^4+275 b^4 a^2-105 b^6\right ) b^3+a \left (75 a^6-449 b^2 a^4+654 b^4 a^2-280 b^6\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin ^2(c+d x) b^3+16 \left (61 a^6-231 b^2 a^4+275 b^4 a^2-105 b^6\right ) b^3+a \left (75 a^6-449 b^2 a^4+654 b^4 a^2-280 b^6\right ) \sin (c+d x) b^2\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\int \frac {-15 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin (c+d x)^2 b^3+16 \left (61 a^6-231 b^2 a^4+275 b^4 a^2-105 b^6\right ) b^3+a \left (75 a^6-449 b^2 a^4+654 b^4 a^2-280 b^6\right ) \sin (c+d x) b^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {\int \frac {15 \csc (c+d x) \left (b^2 \left (5 a^8-95 b^2 a^6+290 b^4 a^4-312 b^6 a^2+112 b^8\right )-a b^3 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {\csc (c+d x) \left (b^2 \left (5 a^8-95 b^2 a^6+290 b^4 a^4-312 b^6 a^2+112 b^8\right )-a b^3 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {b^2 \left (5 a^8-95 b^2 a^6+290 b^4 a^4-312 b^6 a^2+112 b^8\right )-a b^3 \left (27 a^6-113 b^2 a^4+142 b^4 a^2-56 b^6\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {16 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {64 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \int \csc (c+d x)dx}{a}-\frac {32 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}}{3 a}-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}}{a \left (a^2-b^2\right )}+\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}}{30 a^2 b^2}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {\frac {3 \left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{a d (a+b \sin (c+d x))}+\frac {-\frac {5 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \left (-\frac {8 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {-\frac {15 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\frac {15 \left (-\frac {32 b^3 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d}-\frac {b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {16 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cot (c+d x)}{a d}}{2 a}}{3 a}\right )}{4 a}}{a \left (a^2-b^2\right )}}{30 a^2 b^2}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}\) |
Input:
Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
-1/3*(Cot[c + d*x]*Csc[c + d*x]^2)/(b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(6*b^2*d*(a + b*Sin[c + d*x])) + (7*b*Cot[c + d*x]*C sc[c + d*x]^4)/(30*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x ]^5)/(6*a*d*(a + b*Sin[c + d*x])) + (((-5*(16*a^6 - 77*a^4*b^2 + 103*a^2*b ^4 - 42*b^6)*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d) - (3*((-8*b*(15*a^6 - 67 *a^4*b^2 + 87*a^2*b^4 - 35*b^6)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - (-1 /2*((15*((-32*b^3*(2*a^2 - 7*b^2)*(a^2 - b^2)^(5/2)*ArcTan[(2*b + 2*a*Tan[ (c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*d) - (b^2*(5*a^8 - 95*a^6*b^2 + 290 *a^4*b^4 - 312*a^2*b^6 + 112*b^8)*ArcTanh[Cos[c + d*x]])/(a*d)))/a - (16*b ^3*(61*a^6 - 231*a^4*b^2 + 275*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(a*d))/a - (15*b^2*(27*a^6 - 113*a^4*b^2 + 142*a^2*b^4 - 56*b^6)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d))/(3*a)))/(4*a))/(a*(a^2 - b^2)) + (3*(5*a^4 - 20*a^2*b^2 + 14*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(a*d*(a + b*Sin[c + d*x])))/(30*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[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 = 2.34 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{5}}{6}-\frac {4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{5}}{2}+3 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {28 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {32 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {15 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-48 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+40 a \,b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-88 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-192 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{7}}+\frac {4 b \left (\frac {\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\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}+\frac {\left (2 a^{6}-11 a^{4} b^{2}+16 a^{2} b^{4}-7 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}}-\frac {1}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+12 b^{2}}{256 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-96 a^{2} b^{2}+80 b^{4}}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+360 a^{4} b^{2}-800 a^{2} b^{4}+448 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{8}}+\frac {b}{80 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-8 b^{2}\right )}{48 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-36 a^{2} b^{2}+24 b^{4}\right )}{8 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(572\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{5}}{6}-\frac {4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{4}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{5}}{2}+3 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {28 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {32 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {15 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-48 a^{3} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+40 a \,b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-88 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-192 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{7}}+\frac {4 b \left (\frac {\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\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}+\frac {\left (2 a^{6}-11 a^{4} b^{2}+16 a^{2} b^{4}-7 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}}-\frac {1}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+12 b^{2}}{256 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-96 a^{2} b^{2}+80 b^{4}}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+360 a^{4} b^{2}-800 a^{2} b^{4}+448 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{8}}+\frac {b}{80 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-8 b^{2}\right )}{48 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-36 a^{2} b^{2}+24 b^{4}\right )}{8 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(572\) |
| risch | \(\text {Expression too large to display}\) | \(1296\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/64/a^7*(1/6*tan(1/2*d*x+1/2*c)^6*a^5-4/5*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+3*a^3*b^2*tan(1/2*d*x+1/2*c)^4+28/3*a^4*b*ta n(1/2*d*x+1/2*c)^3-32/3*a^2*b^3*tan(1/2*d*x+1/2*c)^3+15/2*a^5*tan(1/2*d*x+ 1/2*c)^2-48*a^3*b^2*tan(1/2*d*x+1/2*c)^2+40*a*b^4*tan(1/2*d*x+1/2*c)^2-88* a^4*b*tan(1/2*d*x+1/2*c)+288*tan(1/2*d*x+1/2*c)*a^2*b^3-192*tan(1/2*d*x+1/ 2*c)*b^5)+4/a^8*b*((1/2*b^2*(a^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)+1/2*a^5 *b-a^3*b^3+1/2*a*b^5)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/ 2*(2*a^6-11*a^4*b^2+16*a^2*b^4-7*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)))-1/384/a^2/tan(1/2*d*x+1/2*c)^6-1/256 *(-6*a^2+12*b^2)/a^4/tan(1/2*d*x+1/2*c)^4-1/128/a^6*(15*a^4-96*a^2*b^2+80* b^4)/tan(1/2*d*x+1/2*c)^2+1/64/a^8*(-20*a^6+360*a^4*b^2-800*a^2*b^4+448*b^ 6)*ln(tan(1/2*d*x+1/2*c))+1/80/a^3*b/tan(1/2*d*x+1/2*c)^5-1/48/a^5*b*(7*a^ 2-8*b^2)/tan(1/2*d*x+1/2*c)^3+1/8*b*(11*a^4-36*a^2*b^2+24*b^4)/a^7/tan(1/2 *d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 1252 vs. \(2 (455) = 910\).
Time = 0.65 (sec) , antiderivative size = 2588, normalized size of antiderivative = 5.39 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
[1/480*(32*(61*a^5*b^2 - 170*a^3*b^4 + 105*a*b^6)*cos(d*x + c)^7 + 2*(165* a^7 - 3386*a^5*b^2 + 8440*a^3*b^4 - 5040*a*b^6)*cos(d*x + c)^5 - 80*(5*a^7 - 94*a^5*b^2 + 218*a^3*b^4 - 126*a*b^6)*cos(d*x + c)^3 + 240*((2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^6 - 2*a^5*b + 9*a^3*b^3 - 7*a*b^5 - 3*(2 *a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^4 + 3*(2*a^5*b - 9*a^3*b^3 + 7* a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^6 - 2*a^4*b^2 + 9*a^2*b^4 - 7*b^6 - 3*(2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^4 + 3*(2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^2)*sin(d*x + c))*s qrt(-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)) + 30*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c) - 15*(5*a^7 - 90*a^5*b^ 2 + 200*a^3*b^4 - 112*a*b^6 - (5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^ 6)*cos(d*x + c)^6 + 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d *x + c)^4 - 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^ 2 + (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7 - (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 90*a^4*b^3 + 200*a^ 2*b^5 - 112*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(5 *a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6 - (5*a^7 - 90*a^5*b^2 + 200...
\[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cot ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)/(a+b*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**6*csc(c + d*x)/(a + b*sin(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+b*sin(d*x+c))^2,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 = 0.25 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.53 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
-1/1920*(120*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*log(abs(tan(1/2* d*x + 1/2*c)))/a^8 - 3840*(2*a^6*b - 11*a^4*b^3 + 16*a^2*b^5 - 7*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^8) - 3840*(a^4*b^3*tan(1/2*d*x + 1/2 *c) - 2*a^2*b^5*tan(1/2*d*x + 1/2*c) + b^7*tan(1/2*d*x + 1/2*c) + a^5*b^2 - 2*a^3*b^4 + a*b^6)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^8) - (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^9*b*tan(1/2*d*x + 1/2*c )^5 - 45*a^10*tan(1/2*d*x + 1/2*c)^4 + 90*a^8*b^2*tan(1/2*d*x + 1/2*c)^4 + 280*a^9*b*tan(1/2*d*x + 1/2*c)^3 - 320*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 + 2 25*a^10*tan(1/2*d*x + 1/2*c)^2 - 1440*a^8*b^2*tan(1/2*d*x + 1/2*c)^2 + 120 0*a^6*b^4*tan(1/2*d*x + 1/2*c)^2 - 2640*a^9*b*tan(1/2*d*x + 1/2*c) + 8640* a^7*b^3*tan(1/2*d*x + 1/2*c) - 5760*a^5*b^5*tan(1/2*d*x + 1/2*c))/a^12 - ( 1470*a^6*tan(1/2*d*x + 1/2*c)^6 - 26460*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 5 8800*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 32928*b^6*tan(1/2*d*x + 1/2*c)^6 + 2 640*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 8640*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 + 5 760*a*b^5*tan(1/2*d*x + 1/2*c)^5 - 225*a^6*tan(1/2*d*x + 1/2*c)^4 + 1440*a ^4*b^2*tan(1/2*d*x + 1/2*c)^4 - 1200*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 280* a^5*b*tan(1/2*d*x + 1/2*c)^3 + 320*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 - 90*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^5*b*tan (1/2*d*x + 1/2*c) - 5*a^6)/(a^8*tan(1/2*d*x + 1/2*c)^6))/d
Time = 23.46 (sec) , antiderivative size = 1810, normalized size of antiderivative = 3.77 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^6/(sin(c + d*x)*(a + b*sin(c + d*x))^2),x)
Output:
tan(c/2 + (d*x)/2)^6/(384*a^2*d) - (tan(c/2 + (d*x)/2)^4*((128*a^2 + 256*b ^2)/(16384*a^4) + 1/(64*a^2) - b^2/(16*a^4)))/d + (tan(c/2 + (d*x)/2)^2*(3 /(128*a^2) + b^2/(8*a^4) - (2*b*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/(1 6*a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4))) /a))/a + ((128*a^2 + 256*b^2)*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/(128*a^2)))/d + (tan(c/2 + (d*x)/2)*(b/(16*a^3) - ((128*a ^2 + 256*b^2)*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/(16*a^3) + (4*b*((12 8*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/a))/(64*a^2) + (4 *b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/a - (4*b*( 3/(64*a^2) + b^2/(4*a^4) - (4*b*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/(1 6*a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4))) /a))/a + ((128*a^2 + 256*b^2)*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/(64*a^2)))/a))/d + (tan(c/2 + (d*x)/2)^3*((b*(128*a^2 + 2 56*b^2))/(3072*a^5) - b/(48*a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/(3*a)))/d - (tan(c/2 + (d*x)/2)^3*((83*a^5*b)/1 5 - (14*a^3*b^3)/3) + a^6/6 + tan(c/2 + (d*x)/2)^4*(6*a^6 + (56*a^2*b^4)/3 - (79*a^4*b^2)/3) - tan(c/2 + (d*x)/2)^5*(112*a*b^5 + (191*a^5*b)/3 - (54 4*a^3*b^3)/3) - tan(c/2 + (d*x)/2)^2*((4*a^6)/3 - (7*a^4*b^2)/5) + tan(c/2 + (d*x)/2)^6*((15*a^6)/2 - 512*b^6 + 872*a^2*b^4 - 352*a^4*b^2) - (8*tan( c/2 + (d*x)/2)^7*(11*a^6*b + 16*b^7 - 8*a^2*b^5 - 20*a^4*b^3))/a - (7*a...
Time = 0.20 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.75 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)/(a+b*sin(d*x+c))^2,x)
Output:
(960*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*si n(c + d*x)**7*a**4*b**2 - 4320*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**4 + 3360*sqrt(a**2 - b**2) *atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**7*b**6 + 9 60*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 - 4320*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**3 + 3360*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**5 + 976* cos(c + d*x)*sin(c + d*x)**6*a**5*b**2 - 2720*cos(c + d*x)*sin(c + d*x)**6 *a**3*b**4 + 1680*cos(c + d*x)*sin(c + d*x)**6*a*b**6 + 571*cos(c + d*x)*s in(c + d*x)**5*a**6*b - 1430*cos(c + d*x)*sin(c + d*x)**5*a**4*b**3 + 840* cos(c + d*x)*sin(c + d*x)**5*a**2*b**5 - 165*cos(c + d*x)*sin(c + d*x)**4* a**7 + 458*cos(c + d*x)*sin(c + d*x)**4*a**5*b**2 - 280*cos(c + d*x)*sin(c + d*x)**4*a**3*b**4 - 222*cos(c + d*x)*sin(c + d*x)**3*a**6*b + 140*cos(c + d*x)*sin(c + d*x)**3*a**4*b**3 + 130*cos(c + d*x)*sin(c + d*x)**2*a**7 - 84*cos(c + d*x)*sin(c + d*x)**2*a**5*b**2 + 56*cos(c + d*x)*sin(c + d*x) *a**6*b - 40*cos(c + d*x)*a**7 - 75*log(tan((c + d*x)/2))*sin(c + d*x)**7* a**6*b + 1350*log(tan((c + d*x)/2))*sin(c + d*x)**7*a**4*b**3 - 3000*log(t an((c + d*x)/2))*sin(c + d*x)**7*a**2*b**5 + 1680*log(tan((c + d*x)/2))*si n(c + d*x)**7*b**7 - 75*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**7 + 13...