Integrand size = 29, antiderivative size = 212 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a \left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{15 d}+\frac {a \left (a^2+6 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{120 d}-\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d} \] Output:
1/16*a*(a^2+6*b^2)*arctanh(cos(d*x+c))/d+1/15*b*(6*a^2+5*b^2)*cot(d*x+c)/d +1/16*a*(a^2+6*b^2)*cot(d*x+c)*csc(d*x+c)/d+1/15*b*(3*a^2-b^2)*cot(d*x+c)* csc(d*x+c)^2/d+1/120*a*(5*a^2-6*b^2)*cot(d*x+c)*csc(d*x+c)^3/d-1/10*b*cot( d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+b* sin(d*x+c))^3/d
Time = 3.55 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {-64 \left (6 a^2 b+5 b^3\right ) \cot \left (\frac {1}{2} (c+d x)\right )-30 \left (a^3+6 a b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-720 a b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+720 a b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+180 a b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-90 a b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )-5 a^3 \sec ^6\left (\frac {1}{2} (c+d x)\right )+96 a^2 b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-640 b^3 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+18 b \sin (c+d x))+2 b \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (45 a b+\left (-3 a^2+20 b^2\right ) \sin (c+d x)\right )+384 a^2 b \tan \left (\frac {1}{2} (c+d x)\right )+320 b^3 \tan \left (\frac {1}{2} (c+d x)\right )-36 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \] Input:
Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
-1/1920*(-64*(6*a^2*b + 5*b^3)*Cot[(c + d*x)/2] - 30*(a^3 + 6*a*b^2)*Csc[( c + d*x)/2]^2 - 120*a^3*Log[Cos[(c + d*x)/2]] - 720*a*b^2*Log[Cos[(c + d*x )/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 720*a*b^2*Log[Sin[(c + d*x)/2]] + 30*a^3*Sec[(c + d*x)/2]^2 + 180*a*b^2*Sec[(c + d*x)/2]^2 - 90*a*b^2*Sec[(c + d*x)/2]^4 - 5*a^3*Sec[(c + d*x)/2]^6 + 96*a^2*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 640*b^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + a^2*Csc[(c + d*x) /2]^6*(5*a + 18*b*Sin[c + d*x]) + 2*b*Csc[(c + d*x)/2]^4*(45*a*b + (-3*a^2 + 20*b^2)*Sin[c + d*x]) + 384*a^2*b*Tan[(c + d*x)/2] + 320*b^3*Tan[(c + d *x)/2] - 36*a^2*b*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/d
Time = 1.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.02, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3368, 3042, 3527, 3042, 3526, 25, 3042, 3510, 25, 3042, 3500, 27, 3042, 3227, 3042, 4254, 24, 4255, 3042, 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.
\(\displaystyle \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^2 (a+b \sin (c+d x))^3}{\sin (c+d x)^7}dx\) |
\(\Big \downarrow \) 3368 |
\(\displaystyle \int \left (1-\sin ^2(c+d x)\right ) \csc ^7(c+d x) (a+b \sin (c+d x))^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (1-\sin (c+d x)^2\right ) (a+b \sin (c+d x))^3}{\sin (c+d x)^7}dx\) |
\(\Big \downarrow \) 3527 |
\(\displaystyle \frac {1}{6} \int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (-4 b \sin ^2(c+d x)-a \sin (c+d x)+3 b\right )dx-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \frac {(a+b \sin (c+d x))^2 \left (-4 b \sin (c+d x)^2-a \sin (c+d x)+3 b\right )}{\sin (c+d x)^6}dx-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int -\csc ^5(c+d x) (a+b \sin (c+d x)) \left (5 a^2+13 b \sin (c+d x) a-6 b^2+14 b^2 \sin ^2(c+d x)\right )dx-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (-\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (5 a^2+13 b \sin (c+d x) a-6 b^2+14 b^2 \sin ^2(c+d x)\right )dx-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (-\frac {1}{5} \int \frac {(a+b \sin (c+d x)) \left (5 a^2+13 b \sin (c+d x) a-6 b^2+14 b^2 \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3510 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int -\csc ^4(c+d x) \left (56 \sin ^2(c+d x) b^3+24 \left (3 a^2-b^2\right ) b+15 a \left (a^2+6 b^2\right ) \sin (c+d x)\right )dx+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \int \csc ^4(c+d x) \left (56 \sin ^2(c+d x) b^3+24 \left (3 a^2-b^2\right ) b+15 a \left (a^2+6 b^2\right ) \sin (c+d x)\right )dx\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \int \frac {56 \sin (c+d x)^2 b^3+24 \left (3 a^2-b^2\right ) b+15 a \left (a^2+6 b^2\right ) \sin (c+d x)}{\sin (c+d x)^4}dx\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\frac {1}{3} \int 3 \csc ^3(c+d x) \left (15 a \left (a^2+6 b^2\right )+8 b \left (6 a^2+5 b^2\right ) \sin (c+d x)\right )dx\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \csc ^3(c+d x) \left (15 a \left (a^2+6 b^2\right )+8 b \left (6 a^2+5 b^2\right ) \sin (c+d x)\right )dx\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \frac {15 a \left (a^2+6 b^2\right )+8 b \left (6 a^2+5 b^2\right ) \sin (c+d x)}{\sin (c+d x)^3}dx\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-15 a \left (a^2+6 b^2\right ) \int \csc ^3(c+d x)dx-8 b \left (6 a^2+5 b^2\right ) \int \csc ^2(c+d x)dx+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-8 b \left (6 a^2+5 b^2\right ) \int \csc (c+d x)^2dx-15 a \left (a^2+6 b^2\right ) \int \csc (c+d x)^3dx+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (6 a^2+5 b^2\right ) \int 1d\cot (c+d x)}{d}-15 a \left (a^2+6 b^2\right ) \int \csc (c+d x)^3dx+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-15 a \left (a^2+6 b^2\right ) \int \csc (c+d x)^3dx+\frac {8 b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-15 a \left (a^2+6 b^2\right ) \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-15 a \left (a^2+6 b^2\right ) \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-15 a \left (a^2+6 b^2\right ) \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2+5 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (3 a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
Input:
Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
-1/6*(Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/d + (((a*(5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + ((8*b*(6*a^2 + 5*b^2)*Cot[c + d*x])/d + (8*b*(3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/d - 15*a*(a^2 + 6*b^2)*(-1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d) ))/4)/5 - (3*b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(5*d))/ 6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.64 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{3}}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \cos \left (d x +c \right )^{3}}{15 \sin \left (d x +c \right )^{3}}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {b^{3} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(224\) |
default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{3}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{3}}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \cos \left (d x +c \right )^{3}}{15 \sin \left (d x +c \right )^{3}}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{3}}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{3}}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {b^{3} \cos \left (d x +c \right )^{3}}{3 \sin \left (d x +c \right )^{3}}}{d}\) | \(224\) |
risch | \(-\frac {-240 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{3} {\mathrm e}^{11 i \left (d x +c \right )}+90 a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+96 i a^{2} b -800 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-85 a^{3} {\mathrm e}^{9 i \left (d x +c \right )}+450 a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-240 i b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+720 i b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-570 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-540 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+480 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-570 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-540 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-960 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+80 i b^{3}-85 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+450 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-576 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+1440 i a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+15 a^{3} {\mathrm e}^{i \left (d x +c \right )}+90 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}\) | \(413\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/6/sin(d*x+c)^6*cos(d*x+c)^3-1/8/sin(d*x+c)^4*cos(d*x+c)^3-1/1 6/sin(d*x+c)^2*cos(d*x+c)^3-1/16*cos(d*x+c)-1/16*ln(csc(d*x+c)-cot(d*x+c)) )+3*a^2*b*(-1/5/sin(d*x+c)^5*cos(d*x+c)^3-2/15/sin(d*x+c)^3*cos(d*x+c)^3)+ 3*a*b^2*(-1/4/sin(d*x+c)^4*cos(d*x+c)^3-1/8/sin(d*x+c)^2*cos(d*x+c)^3-1/8* cos(d*x+c)-1/8*ln(csc(d*x+c)-cot(d*x+c)))-1/3*b^3/sin(d*x+c)^3*cos(d*x+c)^ 3)
Time = 0.09 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.46 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {80 \, a^{3} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 30 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right ) + 15 \, {\left ({\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 6 \, a b^{2} + 3 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left ({\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 6 \, a b^{2} + 3 \, {\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left ({\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 5 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/480*(80*a^3*cos(d*x + c)^3 - 30*(a^3 + 6*a*b^2)*cos(d*x + c)^5 + 30*(a^3 + 6*a*b^2)*cos(d*x + c) + 15*((a^3 + 6*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 6 *a*b^2)*cos(d*x + c)^4 - a^3 - 6*a*b^2 + 3*(a^3 + 6*a*b^2)*cos(d*x + c)^2) *log(1/2*cos(d*x + c) + 1/2) - 15*((a^3 + 6*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 6*a*b^2)*cos(d*x + c)^4 - a^3 - 6*a*b^2 + 3*(a^3 + 6*a*b^2)*cos(d*x + c )^2)*log(-1/2*cos(d*x + c) + 1/2) - 32*((6*a^2*b + 5*b^3)*cos(d*x + c)^5 - 5*(3*a^2*b + b^3)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*c os(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**5*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 90 \, a b^{2} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {160 \, b^{3}}{\tan \left (d x + c\right )^{3}} + \frac {96 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{2} b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
-1/480*(5*a^3*(2*(3*cos(d*x + c)^5 - 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(c os(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) + 90*a*b^2*(2*(cos(d*x + c)^3 + cos(d* x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 160*b^3/tan(d*x + c)^3 + 96*(5*tan(d*x + c)^2 + 3 )*a^2*b/tan(d*x + c)^5)/d
Time = 0.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.67 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 60 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {294 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1764 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 60 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 90 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 1 5*a^3*tan(1/2*d*x + 1/2*c)^4 + 90*a*b^2*tan(1/2*d*x + 1/2*c)^4 + 60*a^2*b* tan(1/2*d*x + 1/2*c)^3 + 80*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d* x + 1/2*c)^2 - 360*a^2*b*tan(1/2*d*x + 1/2*c) - 240*b^3*tan(1/2*d*x + 1/2* c) - 120*(a^3 + 6*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) + (294*a^3*tan(1/2 *d*x + 1/2*c)^6 + 1764*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 360*a^2*b*tan(1/2*d* x + 1/2*c)^5 + 240*b^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x + 1/2*c )^4 - 60*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 80*b^3*tan(1/2*d*x + 1/2*c)^3 - 15 *a^3*tan(1/2*d*x + 1/2*c)^2 - 90*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 36*a^2*b*t an(1/2*d*x + 1/2*c) - 5*a^3)/tan(1/2*d*x + 1/2*c)^6)/d
Time = 34.44 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.38 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{2}+3\,a\,b^2\right )-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^2\,b+\frac {8\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (12\,a^2\,b+8\,b^3\right )+\frac {a^3}{6}+\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}\right )}{64\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{128}+\frac {3\,a\,b^2}{64}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^2\,b}{32}+\frac {b^3}{24}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^3}{16}+\frac {3\,a\,b^2}{8}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2\,b}{16}+\frac {b^3}{8}\right )}{d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d} \] Input:
int((cot(c + d*x)^2*(a + b*sin(c + d*x))^3)/sin(c + d*x)^5,x)
Output:
(a^3*tan(c/2 + (d*x)/2)^6)/(384*d) - (a^3*tan(c/2 + (d*x)/2)^2)/(128*d) - (cot(c/2 + (d*x)/2)^6*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 + a^3/2) - (a^3*tan(c /2 + (d*x)/2)^4)/2 + tan(c/2 + (d*x)/2)^3*(2*a^2*b + (8*b^3)/3) - tan(c/2 + (d*x)/2)^5*(12*a^2*b + 8*b^3) + a^3/6 + (6*a^2*b*tan(c/2 + (d*x)/2))/5)) /(64*d) + (tan(c/2 + (d*x)/2)^4*((3*a*b^2)/64 + a^3/128))/d + (tan(c/2 + ( d*x)/2)^3*((a^2*b)/32 + b^3/24))/d - (log(tan(c/2 + (d*x)/2))*((3*a*b^2)/8 + a^3/16))/d - (tan(c/2 + (d*x)/2)*((3*a^2*b)/16 + b^3/8))/d + (3*a^2*b*t an(c/2 + (d*x)/2)^5)/(160*d)
Time = 0.18 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.16 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {96 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2} b +80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3}+90 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -40 \cos \left (d x +c \right ) a^{3}-15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{3}-90 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a \,b^{2}}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x)
Output:
(96*cos(c + d*x)*sin(c + d*x)**5*a**2*b + 80*cos(c + d*x)*sin(c + d*x)**5* b**3 + 15*cos(c + d*x)*sin(c + d*x)**4*a**3 + 90*cos(c + d*x)*sin(c + d*x) **4*a*b**2 + 48*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 80*cos(c + d*x)*sin( c + d*x)**3*b**3 + 10*cos(c + d*x)*sin(c + d*x)**2*a**3 - 180*cos(c + d*x) *sin(c + d*x)**2*a*b**2 - 144*cos(c + d*x)*sin(c + d*x)*a**2*b - 40*cos(c + d*x)*a**3 - 15*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**3 - 90*log(tan(( c + d*x)/2))*sin(c + d*x)**6*a*b**2)/(240*sin(c + d*x)**6*d)