\(\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1111]

Optimal result

Integrand size = 29, antiderivative size = 261 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {a b \text {arctanh}(\cos (c+d x))}{8 d}-\frac {\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac {\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac {b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac {2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d} \] Output:

-1/8*a*b*arctanh(cos(d*x+c))/d-1/35*(2*a^2+7*b^2)*cot(d*x+c)/d-1/8*a*b*cot 
(d*x+c)*csc(d*x+c)/d-1/105*(3*a^4-18*a^2*b^2+4*b^4)*cot(d*x+c)*csc(d*x+c)^ 
2/a^2/d+1/420*b*(53*a^2-12*b^2)*cot(d*x+c)*csc(d*x+c)^3/a/d+2/35*(4*a^2-b^ 
2)*cot(d*x+c)*csc(d*x+c)^4*(a+b*sin(d*x+c))^2/a^2/d+2/21*b*cot(d*x+c)*csc( 
d*x+c)^5*(a+b*sin(d*x+c))^3/a^2/d-1/7*cot(d*x+c)*csc(d*x+c)^6*(a+b*sin(d*x 
+c))^3/a/d
 

Mathematica [A] (verified)

Time = 2.77 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.23 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\csc ^7(c+d x) \left (840 \left (6 a^2+b^2\right ) \cos (c+d x)+168 \left (14 a^2-b^2\right ) \cos (3 (c+d x))+336 a^2 \cos (5 (c+d x))-504 b^2 \cos (5 (c+d x))-48 a^2 \cos (7 (c+d x))-168 b^2 \cos (7 (c+d x))+3675 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-3675 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+2170 a b \sin (2 (c+d x))-2205 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+2205 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+3080 a b \sin (4 (c+d x))+735 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-735 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+210 a b \sin (6 (c+d x))-105 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+105 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{53760 d} \] Input:

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
 

Output:

-1/53760*(Csc[c + d*x]^7*(840*(6*a^2 + b^2)*Cos[c + d*x] + 168*(14*a^2 - b 
^2)*Cos[3*(c + d*x)] + 336*a^2*Cos[5*(c + d*x)] - 504*b^2*Cos[5*(c + d*x)] 
 - 48*a^2*Cos[7*(c + d*x)] - 168*b^2*Cos[7*(c + d*x)] + 3675*a*b*Log[Cos[( 
c + d*x)/2]]*Sin[c + d*x] - 3675*a*b*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 
2170*a*b*Sin[2*(c + d*x)] - 2205*a*b*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x) 
] + 2205*a*b*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 3080*a*b*Sin[4*(c + 
d*x)] + 735*a*b*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*a*b*Log[Sin[( 
c + d*x)/2]]*Sin[5*(c + d*x)] + 210*a*b*Sin[6*(c + d*x)] - 105*a*b*Log[Cos 
[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*a*b*Log[Sin[(c + d*x)/2]]*Sin[7*(c + 
 d*x)]))/d
 

Rubi [A] (verified)

Time = 1.90 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3372, 27, 3042, 3526, 3042, 3510, 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.

\(\displaystyle \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^2}{\sin (c+d x)^8}dx\)

\(\Big \downarrow \) 3372

\(\displaystyle -\frac {\int 2 \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (-\left (\left (21 a^2-4 b^2\right ) \sin ^2(c+d x)\right )+a b \sin (c+d x)+6 \left (4 a^2-b^2\right )\right )dx}{42 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (-\left (\left (21 a^2-4 b^2\right ) \sin ^2(c+d x)\right )+a b \sin (c+d x)+6 \left (4 a^2-b^2\right )\right )dx}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^2 \left (-\left (\left (21 a^2-4 b^2\right ) \sin (c+d x)^2\right )+a b \sin (c+d x)+6 \left (4 a^2-b^2\right )\right )}{\sin (c+d x)^6}dx}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (-b \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)-a \left (9 a^2-b^2\right ) \sin (c+d x)+b \left (53 a^2-12 b^2\right )\right )dx-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \int \frac {(a+b \sin (c+d x)) \left (-b \left (57 a^2-8 b^2\right ) \sin (c+d x)^2-a \left (9 a^2-b^2\right ) \sin (c+d x)+b \left (53 a^2-12 b^2\right )\right )}{\sin (c+d x)^5}dx-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3510

\(\displaystyle -\frac {\frac {1}{5} \left (-\frac {1}{4} \int \csc ^4(c+d x) \left (105 b \sin (c+d x) a^3+4 b^2 \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)+12 \left (3 a^4-18 b^2 a^2+4 b^4\right )\right )dx-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (-\frac {1}{4} \int \frac {105 b \sin (c+d x) a^3+4 b^2 \left (57 a^2-8 b^2\right ) \sin (c+d x)^2+12 \left (3 a^4-18 b^2 a^2+4 b^4\right )}{\sin (c+d x)^4}dx-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3500

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-\frac {1}{3} \int 9 \csc ^3(c+d x) \left (35 b a^3+4 \left (2 a^2+7 b^2\right ) \sin (c+d x) a^2\right )dx\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \int \csc ^3(c+d x) \left (35 b a^3+4 \left (2 a^2+7 b^2\right ) \sin (c+d x) a^2\right )dx\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \int \frac {35 b a^3+4 \left (2 a^2+7 b^2\right ) \sin (c+d x) a^2}{\sin (c+d x)^3}dx\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3227

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \int \csc ^3(c+d x)dx+4 a^2 \left (2 a^2+7 b^2\right ) \int \csc ^2(c+d x)dx\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \int \csc (c+d x)^3dx+4 a^2 \left (2 a^2+7 b^2\right ) \int \csc (c+d x)^2dx\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \int \csc (c+d x)^3dx-\frac {4 a^2 \left (2 a^2+7 b^2\right ) \int 1d\cot (c+d x)}{d}\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \int \csc (c+d x)^3dx-\frac {4 a^2 \left (2 a^2+7 b^2\right ) \cot (c+d x)}{d}\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {4 a^2 \left (2 a^2+7 b^2\right ) \cot (c+d x)}{d}\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {4 a^2 \left (2 a^2+7 b^2\right ) \cot (c+d x)}{d}\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}+\frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {4 \left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{d}-3 \left (35 a^3 b \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {4 a^2 \left (2 a^2+7 b^2\right ) \cot (c+d x)}{d}\right )\right )-\frac {a b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {6 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}}{21 a^2}-\frac {\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\)

Input:

Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
 

Output:

(2*b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(21*a^2*d) - (Cot 
[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^3)/(7*a*d) - ((-1/4*(a*b*(53 
*a^2 - 12*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/d + ((4*(3*a^4 - 18*a^2*b^2 + 
4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/d - 3*((-4*a^2*(2*a^2 + 7*b^2)*Cot[c + 
 d*x])/d + 35*a^3*b*(-1/2*ArcTanh[Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + 
d*x])/(2*d))))/4)/5 - (6*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b* 
Sin[c + d*x])^2)/(5*d))/(21*a^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3227

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]
 

rule 3372

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])
 

rule 3500

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]
 

rule 3510

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]
 

rule 3526

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]
 

rule 4254

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]
 

rule 4255

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]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{2} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}}{d}\)	\(163\)
default	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {b^{2} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}}{d}\)	\(163\)
risch	\(\frac {336 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+105 a b \,{\mathrm e}^{13 i \left (d x +c \right )}+1680 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-1848 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+1540 a b \,{\mathrm e}^{11 i \left (d x +c \right )}+672 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3360 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1085 a b \,{\mathrm e}^{9 i \left (d x +c \right )}-2520 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+336 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-48 i a^{2}-168 i b^{2}-1085 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+1680 i b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+3360 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1540 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+1680 i a^{2} {\mathrm e}^{10 i \left (d x +c \right )}-840 i b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-105 a b \,{\mathrm e}^{i \left (d x +c \right )}}{420 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\)	\(314\)

Input:

int(cot(d*x+c)^4*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^2*(-1/7/sin(d*x+c)^7*cos(d*x+c)^5-2/35/sin(d*x+c)^5*cos(d*x+c)^5)+2 
*a*b*(-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/s 
in(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^2/sin(d*x+c)^5*cos(d*x+c)^5)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.95 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {48 \, {\left (2 \, a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 336 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left (3 \, a b \cos \left (d x + c\right )^{5} + 8 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

-1/1680*(48*(2*a^2 + 7*b^2)*cos(d*x + c)^7 - 336*(a^2 + b^2)*cos(d*x + c)^ 
5 + 105*(a*b*cos(d*x + c)^6 - 3*a*b*cos(d*x + c)^4 + 3*a*b*cos(d*x + c)^2 
- a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 105*(a*b*cos(d*x + c)^6 
- 3*a*b*cos(d*x + c)^4 + 3*a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*x + c) 
 + 1/2)*sin(d*x + c) - 70*(3*a*b*cos(d*x + c)^5 + 8*a*b*cos(d*x + c)^3 - 3 
*a*b*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)*sin(d*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*csc(d*x+c)**4*(a+b*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.51 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {35 \, a b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {336 \, b^{2}}{\tan \left (d x + c\right )^{5}} - \frac {48 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

1/1680*(35*a*b*(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)) - 336*b^2/tan(d*x + c)^5 - 48*(7*tan( 
d*x + c)^2 + 5)*a^2/tan(d*x + c)^7)/d
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.33 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 70 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 84 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 420 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1680 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 840 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {4356 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 840 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 420 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

1/13440*(15*a^2*tan(1/2*d*x + 1/2*c)^7 + 70*a*b*tan(1/2*d*x + 1/2*c)^6 - 2 
1*a^2*tan(1/2*d*x + 1/2*c)^5 + 84*b^2*tan(1/2*d*x + 1/2*c)^5 - 210*a*b*tan 
(1/2*d*x + 1/2*c)^4 - 105*a^2*tan(1/2*d*x + 1/2*c)^3 - 420*b^2*tan(1/2*d*x 
 + 1/2*c)^3 - 210*a*b*tan(1/2*d*x + 1/2*c)^2 + 1680*a*b*log(abs(tan(1/2*d* 
x + 1/2*c))) + 315*a^2*tan(1/2*d*x + 1/2*c) + 840*b^2*tan(1/2*d*x + 1/2*c) 
 - (4356*a*b*tan(1/2*d*x + 1/2*c)^7 + 315*a^2*tan(1/2*d*x + 1/2*c)^6 + 840 
*b^2*tan(1/2*d*x + 1/2*c)^6 - 210*a*b*tan(1/2*d*x + 1/2*c)^5 - 105*a^2*tan 
(1/2*d*x + 1/2*c)^4 - 420*b^2*tan(1/2*d*x + 1/2*c)^4 - 210*a*b*tan(1/2*d*x 
 + 1/2*c)^3 - 21*a^2*tan(1/2*d*x + 1/2*c)^2 + 84*b^2*tan(1/2*d*x + 1/2*c)^ 
2 + 70*a*b*tan(1/2*d*x + 1/2*c) + 15*a^2)/tan(1/2*d*x + 1/2*c)^7)/d
 

Mupad [B] (verification not implemented)

Time = 34.26 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{5}-\frac {4\,b^2}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^2+8\,b^2\right )-\frac {a^2}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+4\,b^2\right )+2\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {2\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}\right )}{128\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2}{128}+\frac {b^2}{16}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^2}{128}+\frac {b^2}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^2}{640}-\frac {b^2}{160}\right )}{d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \] Input:

int((cot(c + d*x)^4*(a + b*sin(c + d*x))^2)/sin(c + d*x)^4,x)
 

Output:

(a^2*tan(c/2 + (d*x)/2)^7)/(896*d) + (cot(c/2 + (d*x)/2)^7*(tan(c/2 + (d*x 
)/2)^2*(a^2/5 - (4*b^2)/5) - tan(c/2 + (d*x)/2)^6*(3*a^2 + 8*b^2) - a^2/7 
+ tan(c/2 + (d*x)/2)^4*(a^2 + 4*b^2) + 2*a*b*tan(c/2 + (d*x)/2)^3 + 2*a*b* 
tan(c/2 + (d*x)/2)^5 - (2*a*b*tan(c/2 + (d*x)/2))/3))/(128*d) + (tan(c/2 + 
 (d*x)/2)*((3*a^2)/128 + b^2/16))/d - (tan(c/2 + (d*x)/2)^3*(a^2/128 + b^2 
/32))/d - (tan(c/2 + (d*x)/2)^5*(a^2/640 - b^2/160))/d - (a*b*tan(c/2 + (d 
*x)/2)^2)/(64*d) - (a*b*tan(c/2 + (d*x)/2)^4)/(64*d) + (a*b*tan(c/2 + (d*x 
)/2)^6)/(192*d) + (a*b*log(tan(c/2 + (d*x)/2)))/(8*d)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.82 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}-168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+336 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-280 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -120 \cos \left (d x +c \right ) a^{2}+105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7} a b}{840 \sin \left (d x +c \right )^{7} d} \] Input:

int(cot(d*x+c)^4*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x)
 

Output:

( - 48*cos(c + d*x)*sin(c + d*x)**6*a**2 - 168*cos(c + d*x)*sin(c + d*x)** 
6*b**2 - 105*cos(c + d*x)*sin(c + d*x)**5*a*b - 24*cos(c + d*x)*sin(c + d* 
x)**4*a**2 + 336*cos(c + d*x)*sin(c + d*x)**4*b**2 + 490*cos(c + d*x)*sin( 
c + d*x)**3*a*b + 192*cos(c + d*x)*sin(c + d*x)**2*a**2 - 168*cos(c + d*x) 
*sin(c + d*x)**2*b**2 - 280*cos(c + d*x)*sin(c + d*x)*a*b - 120*cos(c + d* 
x)*a**2 + 105*log(tan((c + d*x)/2))*sin(c + d*x)**7*a*b)/(840*sin(c + d*x) 
**7*d)