Integrand size = 29, antiderivative size = 334 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 a \left (a^2+8 b^2\right ) \text {arctanh}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \] Output:
-3/128*a*(a^2+8*b^2)*arctanh(cos(d*x+c))/d-1/35*b*(6*a^2+7*b^2)*cot(d*x+c) /d-3/128*a*(a^2+8*b^2)*cot(d*x+c)*csc(d*x+c)/d-1/280*b*(24*a^4-25*a^2*b^2+ 4*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^2/d-1/2240*(35*a^4-148*a^2*b^2+24*b^4)*co t(d*x+c)*csc(d*x+c)^3/a/d+3/560*b*(23*a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)^4*( a+b*sin(d*x+c))^2/a^2/d+1/112*(21*a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)^5*(a+b* sin(d*x+c))^3/a^2/d+1/14*b*cot(d*x+c)*csc(d*x+c)^6*(a+b*sin(d*x+c))^4/a^2/ d-1/8*cot(d*x+c)*csc(d*x+c)^7*(a+b*sin(d*x+c))^4/a/d
Time = 3.31 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.80 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {6720 a \left (a^2+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6720 a \left (a^2+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^8(c+d x) \left (35 a \left (671 a^2+248 b^2\right ) \cos (c+d x)+35 \left (333 a^3+104 a b^2\right ) \cos (3 (c+d x))+805 a^3 \cos (5 (c+d x))-11480 a b^2 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))-840 a b^2 \cos (7 (c+d x))+21504 a^2 b \sin (2 (c+d x))+2688 b^3 \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+896 b^3 \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-896 b^3 \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))-448 b^3 \sin (8 (c+d x))\right )}{286720 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
-1/286720*(6720*a*(a^2 + 8*b^2)*Log[Cos[(c + d*x)/2]] - 6720*a*(a^2 + 8*b^ 2)*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^8*(35*a*(671*a^2 + 248*b^2)*Cos[c + d*x] + 35*(333*a^3 + 104*a*b^2)*Cos[3*(c + d*x)] + 805*a^3*Cos[5*(c + d* x)] - 11480*a*b^2*Cos[5*(c + d*x)] - 105*a^3*Cos[7*(c + d*x)] - 840*a*b^2* Cos[7*(c + d*x)] + 21504*a^2*b*Sin[2*(c + d*x)] + 2688*b^3*Sin[2*(c + d*x) ] + 16128*a^2*b*Sin[4*(c + d*x)] + 896*b^3*Sin[4*(c + d*x)] + 3072*a^2*b*S in[6*(c + d*x)] - 896*b^3*Sin[6*(c + d*x)] - 384*a^2*b*Sin[8*(c + d*x)] - 448*b^3*Sin[8*(c + d*x)]))/d
Time = 2.47 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {3042, 3372, 3042, 3526, 27, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^3}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \csc ^7(c+d x) (a+b \sin (c+d x))^3 \left (-8 \left (7 a^2-b^2\right ) \sin ^2(c+d x)+3 a b \sin (c+d x)+3 \left (21 a^2-4 b^2\right )\right )dx}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^3 \left (-8 \left (7 a^2-b^2\right ) \sin (c+d x)^2+3 a b \sin (c+d x)+3 \left (21 a^2-4 b^2\right )\right )}{\sin (c+d x)^7}dx}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{6} \int 3 \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (-2 b \left (35 a^2-4 b^2\right ) \sin ^2(c+d x)-a \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 b \left (23 a^2-4 b^2\right )\right )dx-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (-2 b \left (35 a^2-4 b^2\right ) \sin ^2(c+d x)-a \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 b \left (23 a^2-4 b^2\right )\right )dx-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {(a+b \sin (c+d x))^2 \left (-2 b \left (35 a^2-4 b^2\right ) \sin (c+d x)^2-a \left (7 a^2-2 b^2\right ) \sin (c+d x)+3 b \left (23 a^2-4 b^2\right )\right )}{\sin (c+d x)^6}dx-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \int -\csc ^5(c+d x) (a+b \sin (c+d x)) \left (35 a^4-148 b^2 a^2+b \left (109 a^2-2 b^2\right ) \sin (c+d x) a+24 b^4+4 b^2 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (35 a^4-148 b^2 a^2+b \left (109 a^2-2 b^2\right ) \sin (c+d x) a+24 b^4+4 b^2 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-\frac {1}{5} \int \frac {(a+b \sin (c+d x)) \left (35 a^4-148 b^2 a^2+b \left (109 a^2-2 b^2\right ) \sin (c+d x) a+24 b^4+4 b^2 \left (53 a^2-4 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^5}dx-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int -\csc ^4(c+d x) \left (105 \left (a^2+8 b^2\right ) \sin (c+d x) a^3+16 b^3 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)+24 b \left (24 a^4-25 b^2 a^2+4 b^4\right )\right )dx+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \int \csc ^4(c+d x) \left (105 \left (a^2+8 b^2\right ) \sin (c+d x) a^3+16 b^3 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)+24 b \left (24 a^4-25 b^2 a^2+4 b^4\right )\right )dx\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} \int \frac {105 \left (a^2+8 b^2\right ) \sin (c+d x) a^3+16 b^3 \left (53 a^2-4 b^2\right ) \sin (c+d x)^2+24 b \left (24 a^4-25 b^2 a^2+4 b^4\right )}{\sin (c+d x)^4}dx\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\frac {1}{3} \int 3 \csc ^3(c+d x) \left (105 \left (a^2+8 b^2\right ) a^3+64 b \left (6 a^2+7 b^2\right ) \sin (c+d x) a^2\right )dx\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \csc ^3(c+d x) \left (105 \left (a^2+8 b^2\right ) a^3+64 b \left (6 a^2+7 b^2\right ) \sin (c+d x) a^2\right )dx\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\int \frac {105 \left (a^2+8 b^2\right ) a^3+64 b \left (6 a^2+7 b^2\right ) \sin (c+d x) a^2}{\sin (c+d x)^3}dx\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (-64 a^2 b \left (6 a^2+7 b^2\right ) \int \csc ^2(c+d x)dx-105 a^3 \left (a^2+8 b^2\right ) \int \csc ^3(c+d x)dx+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (-64 a^2 b \left (6 a^2+7 b^2\right ) \int \csc (c+d x)^2dx-105 a^3 \left (a^2+8 b^2\right ) \int \csc (c+d x)^3dx+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {64 a^2 b \left (6 a^2+7 b^2\right ) \int 1d\cot (c+d x)}{d}-105 a^3 \left (a^2+8 b^2\right ) \int \csc (c+d x)^3dx+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (-105 a^3 \left (a^2+8 b^2\right ) \int \csc (c+d x)^3dx+\frac {64 a^2 b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (-105 a^3 \left (a^2+8 b^2\right ) \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {64 a^2 b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (-105 a^3 \left (a^2+8 b^2\right ) \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {64 a^2 b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}\right )+\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {a \left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (\frac {64 a^2 b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{d}+\frac {8 b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-105 a^3 \left (a^2+8 b^2\right ) \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )\right )-\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{2 d}}{56 a^2}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]
Output:
(b*Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^4)/(8*a*d) - (-1/2*((21*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/d + (((a*(35*a^ 4 - 148*a^2*b^2 + 24*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + ((64*a^2*b* (6*a^2 + 7*b^2)*Cot[c + d*x])/d + (8*b*(24*a^4 - 25*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/d - 105*a^3*(a^2 + 8*b^2)*(-1/2*ArcTanh[Cos[c + d* x]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d)))/4)/5 - (3*b*(23*a^2 - 4*b^2)*C ot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(5*d))/2)/(56*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_)]*(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 = 0.74 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(280\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+3 a^{2} b \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(280\) |
| risch | \(\frac {26880 i a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-43008 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-5376 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+26880 i a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}-6144 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+105 a^{3} {\mathrm e}^{i \left (d x +c \right )}-805 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-27776 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-11655 a^{3} {\mathrm e}^{11 i \left (d x +c \right )}-11655 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+105 a^{3} {\mathrm e}^{15 i \left (d x +c \right )}-805 a^{3} {\mathrm e}^{13 i \left (d x +c \right )}-23485 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+896 i b^{3}-23485 a^{3} {\mathrm e}^{9 i \left (d x +c \right )}+768 i a^{2} b +840 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}+31360 i b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-22400 i b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+13440 i b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+840 a \,b^{2} {\mathrm e}^{15 i \left (d x +c \right )}-8680 a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-3640 a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-8680 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+11648 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-3640 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-4480 i b^{3} {\mathrm e}^{14 i \left (d x +c \right )}-2688 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+11480 a \,b^{2} {\mathrm e}^{13 i \left (d x +c \right )}+11480 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{16 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{16 d}\) | \(533\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/8/sin(d*x+c)^8*cos(d*x+c)^5-1/16/sin(d*x+c)^6*cos(d*x+c)^5-1/ 64/sin(d*x+c)^4*cos(d*x+c)^5+1/128/sin(d*x+c)^2*cos(d*x+c)^5+1/128*cos(d*x +c)^3+3/128*cos(d*x+c)+3/128*ln(csc(d*x+c)-cot(d*x+c)))+3*a^2*b*(-1/7/sin( d*x+c)^7*cos(d*x+c)^5-2/35/sin(d*x+c)^5*cos(d*x+c)^5)+3*a*b^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)))- 1/5*b^3/sin(d*x+c)^5*cos(d*x+c)^5)
Time = 0.10 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 70 \, {\left (11 \, a^{3} - 40 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 770 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right ) - 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 7 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/8960*(210*(a^3 + 8*a*b^2)*cos(d*x + c)^7 - 70*(11*a^3 - 40*a*b^2)*cos(d* x + c)^5 - 770*(a^3 + 8*a*b^2)*cos(d*x + c)^3 + 210*(a^3 + 8*a*b^2)*cos(d* x + c) - 105*((a^3 + 8*a*b^2)*cos(d*x + c)^8 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^6 + 6*(a^3 + 8*a*b^2)*cos(d*x + c)^4 + a^3 + 8*a*b^2 - 4*(a^3 + 8*a*b^ 2)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + 105*((a^3 + 8*a*b^2)*cos( d*x + c)^8 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^6 + 6*(a^3 + 8*a*b^2)*cos(d*x + c)^4 + a^3 + 8*a*b^2 - 4*(a^3 + 8*a*b^2)*cos(d*x + c)^2)*log(-1/2*cos(d* x + c) + 1/2) + 256*((6*a^2*b + 7*b^3)*cos(d*x + c)^7 - 7*(3*a^2*b + b^3)* cos(d*x + c)^5)*sin(d*x + c))/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d *cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**5*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.74 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, b^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2} b}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/8960*(35*a^3*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^ 3 + 3*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1 )) + 280*a*b^2*(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)) - 1792*b^3/tan(d*x + c)^5 - 768*(7*ta n(d*x + c)^2 + 5)*a^2*b/tan(d*x + c)^7)/d
Time = 0.22 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.37 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/71680*(35*a^3*tan(1/2*d*x + 1/2*c)^8 + 240*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 560*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 336*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 44 8*b^3*tan(1/2*d*x + 1/2*c)^5 - 280*a^3*tan(1/2*d*x + 1/2*c)^4 - 1680*a*b^2 *tan(1/2*d*x + 1/2*c)^4 - 1680*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 2240*b^3*tan (1/2*d*x + 1/2*c)^3 - 1680*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 5040*a^2*b*tan(1 /2*d*x + 1/2*c) + 4480*b^3*tan(1/2*d*x + 1/2*c) + 1680*(a^3 + 8*a*b^2)*log (abs(tan(1/2*d*x + 1/2*c))) - (4566*a^3*tan(1/2*d*x + 1/2*c)^8 + 36528*a*b ^2*tan(1/2*d*x + 1/2*c)^8 + 5040*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4480*b^3*t an(1/2*d*x + 1/2*c)^7 - 1680*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 1680*a^2*b*tan (1/2*d*x + 1/2*c)^5 - 2240*b^3*tan(1/2*d*x + 1/2*c)^5 - 280*a^3*tan(1/2*d* x + 1/2*c)^4 - 1680*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 336*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 448*b^3*tan(1/2*d*x + 1/2*c)^3 + 560*a*b^2*tan(1/2*d*x + 1/2*c )^2 + 240*a^2*b*tan(1/2*d*x + 1/2*c) + 35*a^3)/tan(1/2*d*x + 1/2*c)^8)/d
Time = 34.00 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.14 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{256}+\frac {3\,a\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a^2\,b}{640}-\frac {b^3}{160}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,a^2\,b}{5}-\frac {8\,b^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (18\,a^2\,b+16\,b^3\right )-\frac {a^3}{8}-\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7}-2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{256\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{128}+\frac {b^3}{16}\right )}{d}-\frac {3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d} \] Input:
int((cot(c + d*x)^4*(a + b*sin(c + d*x))^3)/sin(c + d*x)^5,x)
Output:
(a^3*tan(c/2 + (d*x)/2)^8)/(2048*d) - (tan(c/2 + (d*x)/2)^4*((3*a*b^2)/128 + a^3/256))/d - (tan(c/2 + (d*x)/2)^3*((3*a^2*b)/128 + b^3/32))/d - (tan( c/2 + (d*x)/2)^5*((3*a^2*b)/640 - b^3/160))/d + (log(tan(c/2 + (d*x)/2))*( (3*a*b^2)/16 + (3*a^3)/128))/d + (tan(c/2 + (d*x)/2)^4*(6*a*b^2 + a^3) + t an(c/2 + (d*x)/2)^5*(6*a^2*b + 8*b^3) + tan(c/2 + (d*x)/2)^3*((6*a^2*b)/5 - (8*b^3)/5) - tan(c/2 + (d*x)/2)^7*(18*a^2*b + 16*b^3) - a^3/8 - (6*a^2*b *tan(c/2 + (d*x)/2))/7 - 2*a*b^2*tan(c/2 + (d*x)/2)^2 + 6*a*b^2*tan(c/2 + (d*x)/2)^6)/(256*d*tan(c/2 + (d*x)/2)^8) + (tan(c/2 + (d*x)/2)*((9*a^2*b)/ 128 + b^3/16))/d - (3*a*b^2*tan(c/2 + (d*x)/2)^2)/(128*d) + (a*b^2*tan(c/2 + (d*x)/2)^6)/(128*d) + (3*a^2*b*tan(c/2 + (d*x)/2)^7)/(896*d)
Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.97 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a^{2} b -896 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b^{3}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{3}-840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a \,b^{2}-384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2} b +1792 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}-70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3}+3920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+3072 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -896 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-2240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -560 \cos \left (d x +c \right ) a^{3}+105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{8} a^{3}+840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{8} a \,b^{2}}{4480 \sin \left (d x +c \right )^{8} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+b*sin(d*x+c))^3,x)
Output:
( - 768*cos(c + d*x)*sin(c + d*x)**7*a**2*b - 896*cos(c + d*x)*sin(c + d*x )**7*b**3 - 105*cos(c + d*x)*sin(c + d*x)**6*a**3 - 840*cos(c + d*x)*sin(c + d*x)**6*a*b**2 - 384*cos(c + d*x)*sin(c + d*x)**5*a**2*b + 1792*cos(c + d*x)*sin(c + d*x)**5*b**3 - 70*cos(c + d*x)*sin(c + d*x)**4*a**3 + 3920*c os(c + d*x)*sin(c + d*x)**4*a*b**2 + 3072*cos(c + d*x)*sin(c + d*x)**3*a** 2*b - 896*cos(c + d*x)*sin(c + d*x)**3*b**3 + 840*cos(c + d*x)*sin(c + d*x )**2*a**3 - 2240*cos(c + d*x)*sin(c + d*x)**2*a*b**2 - 1920*cos(c + d*x)*s in(c + d*x)*a**2*b - 560*cos(c + d*x)*a**3 + 105*log(tan((c + d*x)/2))*sin (c + d*x)**8*a**3 + 840*log(tan((c + d*x)/2))*sin(c + d*x)**8*a*b**2)/(448 0*sin(c + d*x)**8*d)