Integrand size = 29, antiderivative size = 236 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a b \cot (c+d x)}{5 d}-\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d} \] Output:
-1/16*(a^2+6*b^2)*arctanh(cos(d*x+c))/d-2/5*a*b*cot(d*x+c)/d-1/240*(15*a^4 -80*a^2*b^2+12*b^4)*cot(d*x+c)*csc(d*x+c)/a^2/d+1/60*b*(13*a^2-2*b^2)*cot( d*x+c)*csc(d*x+c)^2/a/d+1/120*(35*a^2-6*b^2)*cot(d*x+c)*csc(d*x+c)^3*(a+b* sin(d*x+c))^2/a^2/d+1/10*b*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^3/a^2/ d-1/6*cot(d*x+c)*csc(d*x+c)^5*(a+b*sin(d*x+c))^3/a/d
Time = 2.22 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.35 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-384 a b \cot \left (\frac {1}{2} (c+d x)\right )-30 \left (a^2-10 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-720 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+720 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-300 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-30 a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac {1}{2} (c+d x)\right )-1344 a b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+768 a b \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )-a \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+12 b \sin (c+d x))+6 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (5 a^2-5 b^2+14 a b \sin (c+d x)\right )+384 a b \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]
Output:
(-384*a*b*Cot[(c + d*x)/2] - 30*(a^2 - 10*b^2)*Csc[(c + d*x)/2]^2 - 120*a^ 2*Log[Cos[(c + d*x)/2]] - 720*b^2*Log[Cos[(c + d*x)/2]] + 120*a^2*Log[Sin[ (c + d*x)/2]] + 720*b^2*Log[Sin[(c + d*x)/2]] + 30*a^2*Sec[(c + d*x)/2]^2 - 300*b^2*Sec[(c + d*x)/2]^2 - 30*a^2*Sec[(c + d*x)/2]^4 + 30*b^2*Sec[(c + d*x)/2]^4 + 5*a^2*Sec[(c + d*x)/2]^6 - 1344*a*b*Csc[c + d*x]^3*Sin[(c + d *x)/2]^4 + 768*a*b*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - a*Csc[(c + d*x)/2]^ 6*(5*a + 12*b*Sin[c + d*x]) + 6*Csc[(c + d*x)/2]^4*(5*a^2 - 5*b^2 + 14*a*b *Sin[c + d*x]) + 384*a*b*Tan[(c + d*x)/2])/(1920*d)
Time = 1.65 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {3042, 3372, 3042, 3526, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 4254, 24, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^2}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (35 a^2+2 b \sin (c+d x) a-6 b^2-3 \left (10 a^2-b^2\right ) \sin ^2(c+d x)\right )dx}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^2 \left (35 a^2+2 b \sin (c+d x) a-6 b^2-3 \left (10 a^2-b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{4} \int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (-b \left (85 a^2-6 b^2\right ) \sin ^2(c+d x)-a \left (15 a^2-2 b^2\right ) \sin (c+d x)+6 b \left (13 a^2-2 b^2\right )\right )dx-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \int \frac {(a+b \sin (c+d x)) \left (-b \left (85 a^2-6 b^2\right ) \sin (c+d x)^2-a \left (15 a^2-2 b^2\right ) \sin (c+d x)+6 b \left (13 a^2-2 b^2\right )\right )}{\sin (c+d x)^4}dx-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {1}{3} \int 3 \csc ^3(c+d x) \left (15 a^4+48 b \sin (c+d x) a^3-80 b^2 a^2+12 b^4+b^2 \left (85 a^2-6 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\int \csc ^3(c+d x) \left (15 a^4+48 b \sin (c+d x) a^3-80 b^2 a^2+12 b^4+b^2 \left (85 a^2-6 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\int \frac {15 a^4+48 b \sin (c+d x) a^3-80 b^2 a^2+12 b^4+b^2 \left (85 a^2-6 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^3}dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {1}{2} \int 3 \csc ^2(c+d x) \left (32 b a^3+5 \left (a^2+6 b^2\right ) \sin (c+d x) a^2\right )dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \int \csc ^2(c+d x) \left (32 b a^3+5 \left (a^2+6 b^2\right ) \sin (c+d x) a^2\right )dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \int \frac {32 b a^3+5 \left (a^2+6 b^2\right ) \sin (c+d x) a^2}{\sin (c+d x)^2}dx-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (32 a^3 b \int \csc ^2(c+d x)dx+5 a^2 \left (a^2+6 b^2\right ) \int \csc (c+d x)dx\right )-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (32 a^3 b \int \csc (c+d x)^2dx+5 a^2 \left (a^2+6 b^2\right ) \int \csc (c+d x)dx\right )-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (5 a^2 \left (a^2+6 b^2\right ) \int \csc (c+d x)dx-\frac {32 a^3 b \int 1d\cot (c+d x)}{d}\right )-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (5 a^2 \left (a^2+6 b^2\right ) \int \csc (c+d x)dx-\frac {32 a^3 b \cot (c+d x)}{d}\right )-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\frac {1}{4} \left (-\frac {2 a b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{d}+\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}-\frac {3}{2} \left (-\frac {32 a^3 b \cot (c+d x)}{d}-\frac {5 a^2 \left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{d}\right )\right )-\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}}{30 a^2}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]
Output:
(b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(10*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*a*d) - (((-3*((-5*a^2*(a ^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/d - (32*a^3*b*Cot[c + d*x])/d))/2 + ((1 5*a^4 - 80*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (2*a*b*(13 *a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/d)/4 - ((35*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(4*d))/(30*a^2)
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_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])
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[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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.42 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a b \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a b \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
| risch | \(\frac {96 i a b +15 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-150 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-960 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+235 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+210 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+480 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+390 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-480 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+390 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+960 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+235 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+210 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-96 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+15 a^{2} {\mathrm e}^{i \left (d x +c \right )}-150 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}\) | \(344\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-1/6/sin(d*x+c)^6*cos(d*x+c)^5-1/24/sin(d*x+c)^4*cos(d*x+c)^5+1/ 48/sin(d*x+c)^2*cos(d*x+c)^5+1/48*cos(d*x+c)^3+1/16*cos(d*x+c)+1/16*ln(csc (d*x+c)-cot(d*x+c)))-2/5*a*b/sin(d*x+c)^5*cos(d*x+c)^5+b^2*(-1/4/sin(d*x+c )^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c)^5+1/8*cos(d*x+c)^3+3/8*cos(d* x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {192 \, a b \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 30 \, {\left (a^{2} - 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 80 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right ) - 15 \, {\left ({\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 6 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left ({\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 6 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/480*(192*a*b*cos(d*x + c)^5*sin(d*x + c) + 30*(a^2 - 10*b^2)*cos(d*x + c )^5 + 80*(a^2 + 6*b^2)*cos(d*x + c)^3 - 30*(a^2 + 6*b^2)*cos(d*x + c) - 15 *((a^2 + 6*b^2)*cos(d*x + c)^6 - 3*(a^2 + 6*b^2)*cos(d*x + c)^4 + 3*(a^2 + 6*b^2)*cos(d*x + c)^2 - a^2 - 6*b^2)*log(1/2*cos(d*x + c) + 1/2) + 15*((a ^2 + 6*b^2)*cos(d*x + c)^6 - 3*(a^2 + 6*b^2)*cos(d*x + c)^4 + 3*(a^2 + 6*b ^2)*cos(d*x + c)^2 - a^2 - 6*b^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**3*(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.76 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 30 \, b^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {192 \, a b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/480*(5*a^2*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(co s(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)) - 30*b^2*(2*(5*cos(d*x + c)^3 - 3*cos(d *x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 192*a*b/tan(d*x + c)^5)/d
Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.31 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, {\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {294 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1764 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a*b*tan(1/2*d*x + 1/2*c)^5 - 15* a^2*tan(1/2*d*x + 1/2*c)^4 + 30*b^2*tan(1/2*d*x + 1/2*c)^4 - 120*a*b*tan(1 /2*d*x + 1/2*c)^3 - 15*a^2*tan(1/2*d*x + 1/2*c)^2 - 240*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*a*b*tan(1/2*d*x + 1/2*c) + 120*(a^2 + 6*b^2)*log(abs(tan(1/ 2*d*x + 1/2*c))) - (294*a^2*tan(1/2*d*x + 1/2*c)^6 + 1764*b^2*tan(1/2*d*x + 1/2*c)^6 + 240*a*b*tan(1/2*d*x + 1/2*c)^5 - 15*a^2*tan(1/2*d*x + 1/2*c)^ 4 - 240*b^2*tan(1/2*d*x + 1/2*c)^4 - 120*a*b*tan(1/2*d*x + 1/2*c)^3 - 15*a ^2*tan(1/2*d*x + 1/2*c)^2 + 30*b^2*tan(1/2*d*x + 1/2*c)^2 + 24*a*b*tan(1/2 *d*x + 1/2*c) + 5*a^2)/tan(1/2*d*x + 1/2*c)^6)/d
Time = 34.13 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^2}{16}+\frac {3\,b^2}{8}\right )}{d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}-b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+8\,b^2\right )-\frac {a^2}{6}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}\right )}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{128}+\frac {b^2}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{128}-\frac {b^2}{64}\right )}{d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{80\,d}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \] Input:
int((cot(c + d*x)^4*(a + b*sin(c + d*x))^2)/sin(c + d*x)^3,x)
Output:
(log(tan(c/2 + (d*x)/2))*(a^2/16 + (3*b^2)/8))/d + (a^2*tan(c/2 + (d*x)/2) ^6)/(384*d) + (cot(c/2 + (d*x)/2)^6*(tan(c/2 + (d*x)/2)^2*(a^2/2 - b^2) + tan(c/2 + (d*x)/2)^4*(a^2/2 + 8*b^2) - a^2/6 + 4*a*b*tan(c/2 + (d*x)/2)^3 - 8*a*b*tan(c/2 + (d*x)/2)^5 - (4*a*b*tan(c/2 + (d*x)/2))/5))/(64*d) - (ta n(c/2 + (d*x)/2)^2*(a^2/128 + b^2/8))/d - (tan(c/2 + (d*x)/2)^4*(a^2/128 - b^2/64))/d - (a*b*tan(c/2 + (d*x)/2)^3)/(16*d) + (a*b*tan(c/2 + (d*x)/2)^ 5)/(80*d) + (a*b*tan(c/2 + (d*x)/2))/(8*d)
Time = 0.18 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.84 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-96 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+150 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-96 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -40 \cos \left (d x +c \right ) a^{2}+15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{2}+90 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} b^{2}}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^2,x)
Output:
( - 96*cos(c + d*x)*sin(c + d*x)**5*a*b - 15*cos(c + d*x)*sin(c + d*x)**4* a**2 + 150*cos(c + d*x)*sin(c + d*x)**4*b**2 + 192*cos(c + d*x)*sin(c + d* x)**3*a*b + 70*cos(c + d*x)*sin(c + d*x)**2*a**2 - 60*cos(c + d*x)*sin(c + d*x)**2*b**2 - 96*cos(c + d*x)*sin(c + d*x)*a*b - 40*cos(c + d*x)*a**2 + 15*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**2 + 90*log(tan((c + d*x)/2))*s in(c + d*x)**6*b**2)/(240*sin(c + d*x)**6*d)