Integrand size = 29, antiderivative size = 275 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=b^3 x-\frac {a \left (a^2+18 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{60 a^2 d}-\frac {\left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d} \] Output:
b^3*x-1/16*a*(a^2+18*b^2)*arctanh(cos(d*x+c))/d-1/60*b*(36*a^4-43*a^2*b^2+ 2*b^4)*cot(d*x+c)/a^2/d-1/240*(15*a^4-84*a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x +c)/a/d+1/120*b*(39*a^2-2*b^2)*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^2/ a^2/d+1/120*(35*a^2-2*b^2)*cot(d*x+c)*csc(d*x+c)^3*(a+b*sin(d*x+c))^3/a^2/ d+1/15*b*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^4/a^2/d-1/6*cot(d*x+c)*c sc(d*x+c)^5*(a+b*sin(d*x+c))^4/a/d
Time = 3.17 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.48 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1920 b^3 c+1920 b^3 d x-64 \left (9 a^2 b-20 b^3\right ) \cot \left (\frac {1}{2} (c+d x)\right )-30 \left (a^3-30 a b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2160 a b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2160 a b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-900 a b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-30 a^3 \sec ^4\left (\frac {1}{2} (c+d x)\right )+90 a b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 a^3 \sec ^6\left (\frac {1}{2} (c+d x)\right )-2016 a^2 b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+640 b^3 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+18 b \sin (c+d x))+2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (15 \left (a^3-3 a b^2\right )+b \left (63 a^2-20 b^2\right ) \sin (c+d x)\right )+576 a^2 b \tan \left (\frac {1}{2} (c+d x)\right )-1280 b^3 \tan \left (\frac {1}{2} (c+d x)\right )+36 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]
Output:
(1920*b^3*c + 1920*b^3*d*x - 64*(9*a^2*b - 20*b^3)*Cot[(c + d*x)/2] - 30*( a^3 - 30*a*b^2)*Csc[(c + d*x)/2]^2 - 120*a^3*Log[Cos[(c + d*x)/2]] - 2160* a*b^2*Log[Cos[(c + d*x)/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 2160*a*b^2*L og[Sin[(c + d*x)/2]] + 30*a^3*Sec[(c + d*x)/2]^2 - 900*a*b^2*Sec[(c + d*x) /2]^2 - 30*a^3*Sec[(c + d*x)/2]^4 + 90*a*b^2*Sec[(c + d*x)/2]^4 + 5*a^3*Se c[(c + d*x)/2]^6 - 2016*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*Csc[(c + d*x)/2]^4*(15*(a^3 - 3*a*b^2) + b*(63*a^2 - 20*b^2 )*Sin[c + d*x]) + 576*a^2*b*Tan[(c + d*x)/2] - 1280*b^3*Tan[(c + d*x)/2] + 36*a^2*b*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(1920*d)
Time = 1.93 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3372, 3042, 3526, 27, 3042, 3526, 25, 3042, 3510, 25, 3042, 3500, 27, 3042, 3214, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^3}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^3 \left (-30 \sin ^2(c+d x) a^2+35 a^2+3 b \sin (c+d x) a-2 b^2\right )dx}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^3 \left (-30 \sin (c+d x)^2 a^2+35 a^2+3 b \sin (c+d x) a-2 b^2\right )}{\sin (c+d x)^5}dx}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{4} \int 3 \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (-40 a^2 b \sin ^2(c+d x)-a \left (5 a^2-2 b^2\right ) \sin (c+d x)+b \left (39 a^2-2 b^2\right )\right )dx-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{4} \int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (-40 a^2 b \sin ^2(c+d x)-a \left (5 a^2-2 b^2\right ) \sin (c+d x)+b \left (39 a^2-2 b^2\right )\right )dx-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \int \frac {(a+b \sin (c+d x))^2 \left (-40 a^2 b \sin (c+d x)^2-a \left (5 a^2-2 b^2\right ) \sin (c+d x)+b \left (39 a^2-2 b^2\right )\right )}{\sin (c+d x)^4}dx-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \int -\csc ^3(c+d x) (a+b \sin (c+d x)) \left (15 a^4-84 b^2 a^2+120 b^2 \sin ^2(c+d x) a^2+b \left (57 a^2-2 b^2\right ) \sin (c+d x) a+4 b^4\right )dx-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3}{4} \left (-\frac {1}{3} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (15 a^4-84 b^2 a^2+120 b^2 \sin ^2(c+d x) a^2+b \left (57 a^2-2 b^2\right ) \sin (c+d x) a+4 b^4\right )dx-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \left (-\frac {1}{3} \int \frac {(a+b \sin (c+d x)) \left (15 a^4-84 b^2 a^2+120 b^2 \sin (c+d x)^2 a^2+b \left (57 a^2-2 b^2\right ) \sin (c+d x) a+4 b^4\right )}{\sin (c+d x)^3}dx-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int -\csc ^2(c+d x) \left (15 \left (a^2+18 b^2\right ) \sin (c+d x) a^3+240 b^3 \sin ^2(c+d x) a^2+4 b \left (36 a^4-43 b^2 a^2+2 b^4\right )\right )dx+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}-\frac {1}{2} \int \csc ^2(c+d x) \left (15 \left (a^2+18 b^2\right ) \sin (c+d x) a^3+240 b^3 \sin ^2(c+d x) a^2+4 b \left (36 a^4-43 b^2 a^2+2 b^4\right )\right )dx\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}-\frac {1}{2} \int \frac {15 \left (a^2+18 b^2\right ) \sin (c+d x) a^3+240 b^3 \sin (c+d x)^2 a^2+4 b \left (36 a^4-43 b^2 a^2+2 b^4\right )}{\sin (c+d x)^2}dx\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-\int 15 \csc (c+d x) \left (\left (a^2+18 b^2\right ) a^3+16 b^3 \sin (c+d x) a^2\right )dx\right )+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-15 \int \csc (c+d x) \left (\left (a^2+18 b^2\right ) a^3+16 b^3 \sin (c+d x) a^2\right )dx\right )+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-15 \int \frac {\left (a^2+18 b^2\right ) a^3+16 b^3 \sin (c+d x) a^2}{\sin (c+d x)}dx\right )+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-15 \left (a^3 \left (a^2+18 b^2\right ) \int \csc (c+d x)dx+16 a^2 b^3 x\right )\right )+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-15 \left (a^3 \left (a^2+18 b^2\right ) \int \csc (c+d x)dx+16 a^2 b^3 x\right )\right )+\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {a \left (15 a^4-84 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} \left (\frac {4 b \left (36 a^4-43 a^2 b^2+2 b^4\right ) \cot (c+d x)}{d}-15 \left (16 a^2 b^3 x-\frac {a^3 \left (a^2+18 b^2\right ) \text {arctanh}(\cos (c+d x))}{d}\right )\right )\right )-\frac {b \left (39 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\right )-\frac {\left (35 a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}}{30 a^2}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]
Output:
(b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(15*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(6*a*d) - (-1/4*((35*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/d + (3*(((-15*( 16*a^2*b^3*x - (a^3*(a^2 + 18*b^2)*ArcTanh[Cos[c + d*x]])/d) + (4*b*(36*a^ 4 - 43*a^2*b^2 + 2*b^4)*Cot[c + d*x])/d)/2 + (a*(15*a^4 - 84*a^2*b^2 + 4*b ^4)*Cot[c + d*x]*Csc[c + d*x])/(2*d))/3 - (b*(39*a^2 - 2*b^2)*Cot[c + d*x] *Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(3*d)))/4)/(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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{2} b \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(227\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{2} b \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(227\) |
| risch | \(b^{3} x +\frac {i \left (-235 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+180 i a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-630 i a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-630 i a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-720 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+480 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-390 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-15 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+720 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-1920 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+450 i a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+450 i a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-1440 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+3200 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+180 i a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-15 i a^{3} {\mathrm e}^{11 i \left (d x +c \right )}+1440 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-2880 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-390 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-235 i a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-144 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+1440 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+144 a^{2} b -320 b^{3}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}\) | \(451\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(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)^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)))-3/5*a^2*b/sin(d*x+c)^5*cos(d*x+c)^5+3*a*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)))+b^3*(-1/3*cot(d*x+c)^3+cot(d*x+c )+d*x+c))
Time = 0.10 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.36 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {480 \, b^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, b^{3} d x \cos \left (d x + c\right )^{4} + 1440 \, b^{3} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (a^{3} - 30 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 480 \, b^{3} d x + 80 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right ) - 15 \, {\left ({\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 18 \, a b^{2} + 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left ({\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - a^{3} - 18 \, a b^{2} + 3 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 32 \, {\left (35 \, b^{3} \cos \left (d x + c\right )^{3} + {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \cos \left (d x + c\right )^{5} - 15 \, b^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/480*(480*b^3*d*x*cos(d*x + c)^6 - 1440*b^3*d*x*cos(d*x + c)^4 + 1440*b^3 *d*x*cos(d*x + c)^2 + 30*(a^3 - 30*a*b^2)*cos(d*x + c)^5 - 480*b^3*d*x + 8 0*(a^3 + 18*a*b^2)*cos(d*x + c)^3 - 30*(a^3 + 18*a*b^2)*cos(d*x + c) - 15* ((a^3 + 18*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 18*a*b^2)*cos(d*x + c)^4 - a^3 - 18*a*b^2 + 3*(a^3 + 18*a*b^2)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/ 2) + 15*((a^3 + 18*a*b^2)*cos(d*x + c)^6 - 3*(a^3 + 18*a*b^2)*cos(d*x + c) ^4 - a^3 - 18*a*b^2 + 3*(a^3 + 18*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) + 32*(35*b^3*cos(d*x + c)^3 + (9*a^2*b - 20*b^3)*cos(d*x + c)^ 5 - 15*b^3*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c )^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**3*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.79 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {160 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} b^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a b^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {288 \, a^{2} b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/480*(160*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*b^3 + 5*a ^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(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*(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)) - 288*a^2*b/tan(d*x + c)^5)/d
Time = 0.21 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.45 \[ \int \cot ^4(c+d x) \csc ^3(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} - 180 \, 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} - 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1920 \, {\left (d x + c\right )} b^{3} + 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1200 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, {\left (a^{3} + 18 \, 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} + 5292 \, 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} - 1200 \, 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} - 720 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, 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)^4*csc(d*x+c)^3*(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 - 180*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 - 720*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 1920*(d*x + c)*b^3 + 36 0*a^2*b*tan(1/2*d*x + 1/2*c) - 1200*b^3*tan(1/2*d*x + 1/2*c) + 120*(a^3 + 18*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 + 5292*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 - 1200*b^3*tan(1/2*d*x + 1/2*c)^5 - 15*a^3*tan(1/2*d*x + 1/2*c)^4 - 720*a*b^ 2*tan(1/2*d*x + 1/2*c)^4 - 180*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*tan(1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^6) /d
Time = 34.30 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.62 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {2\,b^3\,\mathrm {atan}\left (\frac {4\,b^6}{\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}+\frac {9\,a\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6\right )}+\frac {a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {a^3\,b^3}{4}+\frac {9\,a\,b^5}{2}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6\right )}\right )}{d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{128}+\frac {3\,a\,b^2}{8}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a\,b^2}{64}-\frac {a^3}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{32}-\frac {b^3}{24}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a\,b^2-\frac {a^3}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{2}+24\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,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-40\,b^3\right )+\frac {a^3}{6}+\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}}{64\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2\,b}{16}-\frac {5\,b^3}{8}\right )}{d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+18\,b^2\right )}{16\,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)^4*(a + b*sin(c + d*x))^3)/sin(c + d*x)^3,x)
Output:
(2*b^3*atan((4*b^6)/((9*a*b^5)/2 + (a^3*b^3)/4 - 4*b^6*tan(c/2 + (d*x)/2)) + (9*a*b^5*tan(c/2 + (d*x)/2))/(2*((9*a*b^5)/2 + (a^3*b^3)/4 - 4*b^6*tan( c/2 + (d*x)/2))) + (a^3*b^3*tan(c/2 + (d*x)/2))/(4*((9*a*b^5)/2 + (a^3*b^3 )/4 - 4*b^6*tan(c/2 + (d*x)/2)))))/d + (a^3*tan(c/2 + (d*x)/2)^6)/(384*d) - (tan(c/2 + (d*x)/2)^2*((3*a*b^2)/8 + a^3/128))/d + (tan(c/2 + (d*x)/2)^4 *((3*a*b^2)/64 - a^3/128))/d - (tan(c/2 + (d*x)/2)^3*((3*a^2*b)/32 - b^3/2 4))/d - (tan(c/2 + (d*x)/2)^2*(3*a*b^2 - a^3/2) - tan(c/2 + (d*x)/2)^4*(24 *a*b^2 + a^3/2) - tan(c/2 + (d*x)/2)^3*(6*a^2*b - (8*b^3)/3) + tan(c/2 + ( d*x)/2)^5*(12*a^2*b - 40*b^3) + a^3/6 + (6*a^2*b*tan(c/2 + (d*x)/2))/5)/(6 4*d*tan(c/2 + (d*x)/2)^6) + (tan(c/2 + (d*x)/2)*((3*a^2*b)/16 - (5*b^3)/8) )/d + (a*log(tan(c/2 + (d*x)/2))*(a^2 + 18*b^2))/(16*d) + (3*a^2*b*tan(c/2 + (d*x)/2)^5)/(160*d)
Time = 0.18 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.95 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2} b +320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3}+450 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+288 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -40 \cos \left (d x +c \right ) a^{3}+15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a^{3}+270 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6} a \,b^{2}+240 \sin \left (d x +c \right )^{6} b^{3} d x}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+b*sin(d*x+c))^3,x)
Output:
( - 144*cos(c + d*x)*sin(c + d*x)**5*a**2*b + 320*cos(c + d*x)*sin(c + d*x )**5*b**3 - 15*cos(c + d*x)*sin(c + d*x)**4*a**3 + 450*cos(c + d*x)*sin(c + d*x)**4*a*b**2 + 288*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 80*cos(c + d* x)*sin(c + d*x)**3*b**3 + 70*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 + 270*l og(tan((c + d*x)/2))*sin(c + d*x)**6*a*b**2 + 240*sin(c + d*x)**6*b**3*d*x )/(240*sin(c + d*x)**6*d)