\(\int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\) [416]

Optimal result

Integrand size = 21, antiderivative size = 423 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {1}{256} \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) x-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{40320 d}+\frac {\left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \cos (c+d x) \sin (c+d x)}{256 d}-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d} \] Output:

1/256*(128*a^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*x-11/40320*a*b* 
(1792*a^6+10536*a^4*b^2+9588*a^2*b^4+1289*b^6)*cos(d*x+c)^3/d+1/256*(128*a 
^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*cos(d*x+c)*sin(d*x+c)/d-1/1 
3440*b*(6272*a^6+28088*a^4*b^2+15956*a^2*b^4+735*b^6)*cos(d*x+c)^3*(a+b*si 
n(d*x+c))/d-13/3360*a*b*(112*a^4+348*a^2*b^2+101*b^4)*cos(d*x+c)^3*(a+b*si 
n(d*x+c))^2/d-1/2016*b*(784*a^4+1500*a^2*b^2+147*b^4)*cos(d*x+c)^3*(a+b*si 
n(d*x+c))^3/d-1/336*a*b*(112*a^2+109*b^2)*cos(d*x+c)^3*(a+b*sin(d*x+c))^4/ 
d-1/240*b*(64*a^2+21*b^2)*cos(d*x+c)^3*(a+b*sin(d*x+c))^5/d-17/90*a*b*cos( 
d*x+c)^3*(a+b*sin(d*x+c))^6/d-1/10*b*cos(d*x+c)^3*(a+b*sin(d*x+c))^7/d
 

Mathematica [A] (verified)

Time = 2.72 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {322560 a^8 c+2257920 a^6 b^2 c+2822400 a^4 b^4 c+705600 a^2 b^6 c+17640 b^8 c+322560 a^8 d x+2257920 a^6 b^2 d x+2822400 a^4 b^4 d x+705600 a^2 b^6 d x+17640 b^8 d x-40320 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \cos (c+d x)-26880 \left (16 a^7 b+28 a^5 b^3+7 a^3 b^5\right ) \cos (3 (c+d x))+451584 a^5 b^3 \cos (5 (c+d x))+338688 a^3 b^5 \cos (5 (c+d x))+32256 a b^7 \cos (5 (c+d x))-80640 a^3 b^5 \cos (7 (c+d x))-14400 a b^7 \cos (7 (c+d x))+2240 a b^7 \cos (9 (c+d x))+161280 a^8 \sin (2 (c+d x))-705600 a^4 b^4 \sin (2 (c+d x))-282240 a^2 b^6 \sin (2 (c+d x))-8820 b^8 \sin (2 (c+d x))-564480 a^6 b^2 \sin (4 (c+d x))-705600 a^4 b^4 \sin (4 (c+d x))-141120 a^2 b^6 \sin (4 (c+d x))-2520 b^8 \sin (4 (c+d x))+235200 a^4 b^4 \sin (6 (c+d x))+94080 a^2 b^6 \sin (6 (c+d x))+2730 b^8 \sin (6 (c+d x))-17640 a^2 b^6 \sin (8 (c+d x))-945 b^8 \sin (8 (c+d x))+126 b^8 \sin (10 (c+d x))}{645120 d} \] Input:

Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]
 

Output:

(322560*a^8*c + 2257920*a^6*b^2*c + 2822400*a^4*b^4*c + 705600*a^2*b^6*c + 
 17640*b^8*c + 322560*a^8*d*x + 2257920*a^6*b^2*d*x + 2822400*a^4*b^4*d*x 
+ 705600*a^2*b^6*d*x + 17640*b^8*d*x - 40320*a*b*(32*a^6 + 112*a^4*b^2 + 7 
0*a^2*b^4 + 7*b^6)*Cos[c + d*x] - 26880*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5 
)*Cos[3*(c + d*x)] + 451584*a^5*b^3*Cos[5*(c + d*x)] + 338688*a^3*b^5*Cos[ 
5*(c + d*x)] + 32256*a*b^7*Cos[5*(c + d*x)] - 80640*a^3*b^5*Cos[7*(c + d*x 
)] - 14400*a*b^7*Cos[7*(c + d*x)] + 2240*a*b^7*Cos[9*(c + d*x)] + 161280*a 
^8*Sin[2*(c + d*x)] - 705600*a^4*b^4*Sin[2*(c + d*x)] - 282240*a^2*b^6*Sin 
[2*(c + d*x)] - 8820*b^8*Sin[2*(c + d*x)] - 564480*a^6*b^2*Sin[4*(c + d*x) 
] - 705600*a^4*b^4*Sin[4*(c + d*x)] - 141120*a^2*b^6*Sin[4*(c + d*x)] - 25 
20*b^8*Sin[4*(c + d*x)] + 235200*a^4*b^4*Sin[6*(c + d*x)] + 94080*a^2*b^6* 
Sin[6*(c + d*x)] + 2730*b^8*Sin[6*(c + d*x)] - 17640*a^2*b^6*Sin[8*(c + d* 
x)] - 945*b^8*Sin[8*(c + d*x)] + 126*b^8*Sin[10*(c + d*x)])/(645120*d)
 

Rubi [A] (verified)

Time = 2.59 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.01, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.048, Rules used = {3042, 3171, 3042, 3341, 27, 3042, 3341, 27, 3042, 3341, 3042, 3341, 27, 3042, 3341, 3042, 3341, 3042, 3148, 3042, 3115, 24}

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.

\(\displaystyle \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3171

\(\displaystyle \frac {1}{10} \int \cos ^2(c+d x) (a+b \sin (c+d x))^6 \left (10 a^2+17 b \sin (c+d x) a+7 b^2\right )dx-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \int \cos (c+d x)^2 (a+b \sin (c+d x))^6 \left (10 a^2+17 b \sin (c+d x) a+7 b^2\right )dx-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{9} \int 3 \cos ^2(c+d x) (a+b \sin (c+d x))^5 \left (5 a \left (6 a^2+11 b^2\right )+b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right )dx-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \int \cos ^2(c+d x) (a+b \sin (c+d x))^5 \left (5 a \left (6 a^2+11 b^2\right )+b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right )dx-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \int \cos (c+d x)^2 (a+b \sin (c+d x))^5 \left (5 a \left (6 a^2+11 b^2\right )+b \left (64 a^2+21 b^2\right ) \sin (c+d x)\right )dx-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{8} \int 5 \cos ^2(c+d x) (a+b \sin (c+d x))^4 \left (48 a^4+152 b^2 a^2+b \left (112 a^2+109 b^2\right ) \sin (c+d x) a+21 b^4\right )dx-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \int \cos ^2(c+d x) (a+b \sin (c+d x))^4 \left (48 a^4+152 b^2 a^2+b \left (112 a^2+109 b^2\right ) \sin (c+d x) a+21 b^4\right )dx-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \int \cos (c+d x)^2 (a+b \sin (c+d x))^4 \left (48 a^4+152 b^2 a^2+b \left (112 a^2+109 b^2\right ) \sin (c+d x) a+21 b^4\right )dx-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \left (a \left (336 a^4+1512 b^2 a^2+583 b^4\right )+b \left (784 a^4+1500 b^2 a^2+147 b^4\right ) \sin (c+d x)\right )dx-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \int \cos (c+d x)^2 (a+b \sin (c+d x))^3 \left (a \left (336 a^4+1512 b^2 a^2+583 b^4\right )+b \left (784 a^4+1500 b^2 a^2+147 b^4\right ) \sin (c+d x)\right )dx-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{6} \int 3 \cos ^2(c+d x) (a+b \sin (c+d x))^2 \left (672 a^6+3808 b^2 a^4+2666 b^4 a^2+13 b \left (112 a^4+348 b^2 a^2+101 b^4\right ) \sin (c+d x) a+147 b^6\right )dx-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \left (672 a^6+3808 b^2 a^4+2666 b^4 a^2+13 b \left (112 a^4+348 b^2 a^2+101 b^4\right ) \sin (c+d x) a+147 b^6\right )dx-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \int \cos (c+d x)^2 (a+b \sin (c+d x))^2 \left (672 a^6+3808 b^2 a^4+2666 b^4 a^2+13 b \left (112 a^4+348 b^2 a^2+101 b^4\right ) \sin (c+d x) a+147 b^6\right )dx-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (a \left (3360 a^6+21952 b^2 a^4+22378 b^4 a^2+3361 b^6\right )+b \left (6272 a^6+28088 b^2 a^4+15956 b^4 a^2+735 b^6\right ) \sin (c+d x)\right )dx-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \int \cos (c+d x)^2 (a+b \sin (c+d x)) \left (a \left (3360 a^6+21952 b^2 a^4+22378 b^4 a^2+3361 b^6\right )+b \left (6272 a^6+28088 b^2 a^4+15956 b^4 a^2+735 b^6\right ) \sin (c+d x)\right )dx-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (105 \left (128 a^8+896 b^2 a^6+1120 b^4 a^4+280 b^6 a^2+7 b^8\right )+11 a b \left (1792 a^6+10536 b^2 a^4+9588 b^4 a^2+1289 b^6\right ) \sin (c+d x)\right )dx-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos (c+d x)^2 \left (105 \left (128 a^8+896 b^2 a^6+1120 b^4 a^4+280 b^6 a^2+7 b^8\right )+11 a b \left (1792 a^6+10536 b^2 a^4+9588 b^4 a^2+1289 b^6\right ) \sin (c+d x)\right )dx-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3148

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int \cos ^2(c+d x)dx-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {5}{8} \left (\frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {11 a b \left (1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {b \left (6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {13 a b \left (112 a^4+348 a^2 b^2+101 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {b \left (784 a^4+1500 a^2 b^2+147 b^4\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\right )-\frac {a b \left (112 a^2+109 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{7 d}\right )-\frac {b \left (64 a^2+21 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{8 d}\right )-\frac {17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{9 d}\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}\)

Input:

Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]
 

Output:

-1/10*(b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^7)/d + ((-17*a*b*Cos[c + d*x] 
^3*(a + b*Sin[c + d*x])^6)/(9*d) + (-1/8*(b*(64*a^2 + 21*b^2)*Cos[c + d*x] 
^3*(a + b*Sin[c + d*x])^5)/d + (5*(-1/7*(a*b*(112*a^2 + 109*b^2)*Cos[c + d 
*x]^3*(a + b*Sin[c + d*x])^4)/d + (-1/6*(b*(784*a^4 + 1500*a^2*b^2 + 147*b 
^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/d + ((-13*a*b*(112*a^4 + 348*a^ 
2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(5*d) + (-1/4*(b*( 
6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*(a + b* 
Sin[c + d*x]))/d + ((-11*a*b*(1792*a^6 + 10536*a^4*b^2 + 9588*a^2*b^4 + 12 
89*b^6)*Cos[c + d*x]^3)/(3*d) + 105*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 
+ 280*a^2*b^6 + 7*b^8)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)/5)/2) 
/7))/8)/3)/10
 

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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3148

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + 
 Simp[a   Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && 
 (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
 

rule 3171

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p)   Int[(g*Cos[e + f*x])^p* 
(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) 
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] 
 && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
 

rule 3341

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* 
(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S 
imp[1/(m + p + 1)   Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim 
p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] 
 &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S 
implerQ[c + d*x, a + b*x])
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.17

\[\frac {a^{8} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 a^{7} b \cos \left (d x +c \right )^{3}}{3}+28 a^{6} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+56 a^{5} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+70 a^{4} b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+56 a^{3} b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )+28 b^{6} a^{2} \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{3}}{8}-\frac {5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{48}-\frac {5 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{64}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+8 a \,b^{7} \left (-\frac {\sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{3}}{9}-\frac {2 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{3}}{21}-\frac {8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{105}-\frac {16 \cos \left (d x +c \right )^{3}}{315}\right )+b^{8} \left (-\frac {\sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{3}}{10}-\frac {7 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{3}}{80}-\frac {7 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{96}-\frac {7 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{128}+\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}\]

Input:

int(cos(d*x+c)^2*(a+b*sin(d*x+c))^8,x)
 

Output:

1/d*(a^8*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)-8/3*a^7*b*cos(d*x+c)^3+ 
28*a^6*b^2*(-1/4*sin(d*x+c)*cos(d*x+c)^3+1/8*cos(d*x+c)*sin(d*x+c)+1/8*d*x 
+1/8*c)+56*a^5*b^3*(-1/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+70*a 
^4*b^4*(-1/6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*sin(d*x+c)*cos(d*x+c)^3+1/16*co 
s(d*x+c)*sin(d*x+c)+1/16*d*x+1/16*c)+56*a^3*b^5*(-1/7*sin(d*x+c)^4*cos(d*x 
+c)^3-4/35*sin(d*x+c)^2*cos(d*x+c)^3-8/105*cos(d*x+c)^3)+28*b^6*a^2*(-1/8* 
sin(d*x+c)^5*cos(d*x+c)^3-5/48*sin(d*x+c)^3*cos(d*x+c)^3-5/64*sin(d*x+c)*c 
os(d*x+c)^3+5/128*cos(d*x+c)*sin(d*x+c)+5/128*d*x+5/128*c)+8*a*b^7*(-1/9*s 
in(d*x+c)^6*cos(d*x+c)^3-2/21*sin(d*x+c)^4*cos(d*x+c)^3-8/105*sin(d*x+c)^2 
*cos(d*x+c)^3-16/315*cos(d*x+c)^3)+b^8*(-1/10*sin(d*x+c)^7*cos(d*x+c)^3-7/ 
80*sin(d*x+c)^5*cos(d*x+c)^3-7/96*sin(d*x+c)^3*cos(d*x+c)^3-7/128*sin(d*x+ 
c)*cos(d*x+c)^3+7/256*cos(d*x+c)*sin(d*x+c)+7/256*d*x+7/256*c))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {71680 \, a b^{7} \cos \left (d x + c\right )^{9} - 92160 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{7} + 129024 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{5} - 215040 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} + 315 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} d x + 21 \, {\left (384 \, b^{8} \cos \left (d x + c\right )^{9} - 48 \, {\left (280 \, a^{2} b^{6} + 31 \, b^{8}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (5600 \, a^{4} b^{4} + 4760 \, a^{2} b^{6} + 263 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (2688 \, a^{6} b^{2} + 7840 \, a^{4} b^{4} + 3304 \, a^{2} b^{6} + 121 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

1/80640*(71680*a*b^7*cos(d*x + c)^9 - 92160*(7*a^3*b^5 + 3*a*b^7)*cos(d*x 
+ c)^7 + 129024*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^5 - 215040 
*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^3 + 315*(128*a^8 + 8 
96*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*d*x + 21*(384*b^8*cos(d*x 
 + c)^9 - 48*(280*a^2*b^6 + 31*b^8)*cos(d*x + c)^7 + 8*(5600*a^4*b^4 + 476 
0*a^2*b^6 + 263*b^8)*cos(d*x + c)^5 - 10*(2688*a^6*b^2 + 7840*a^4*b^4 + 33 
04*a^2*b^6 + 121*b^8)*cos(d*x + c)^3 + 15*(128*a^8 + 896*a^6*b^2 + 1120*a^ 
4*b^4 + 280*a^2*b^6 + 7*b^8)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (415) = 830\).

Time = 1.74 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.64 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**2*(a+b*sin(d*x+c))**8,x)
 

Output:

Piecewise((a**8*x*sin(c + d*x)**2/2 + a**8*x*cos(c + d*x)**2/2 + a**8*sin( 
c + d*x)*cos(c + d*x)/(2*d) - 8*a**7*b*cos(c + d*x)**3/(3*d) + 7*a**6*b**2 
*x*sin(c + d*x)**4/2 + 7*a**6*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2 + 7*a 
**6*b**2*x*cos(c + d*x)**4/2 + 7*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)/(2 
*d) - 7*a**6*b**2*sin(c + d*x)*cos(c + d*x)**3/(2*d) - 56*a**5*b**3*sin(c 
+ d*x)**2*cos(c + d*x)**3/(3*d) - 112*a**5*b**3*cos(c + d*x)**5/(15*d) + 3 
5*a**4*b**4*x*sin(c + d*x)**6/8 + 105*a**4*b**4*x*sin(c + d*x)**4*cos(c + 
d*x)**2/8 + 105*a**4*b**4*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + 35*a**4*b* 
*4*x*cos(c + d*x)**6/8 + 35*a**4*b**4*sin(c + d*x)**5*cos(c + d*x)/(8*d) - 
 35*a**4*b**4*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) - 35*a**4*b**4*sin(c + 
 d*x)*cos(c + d*x)**5/(8*d) - 56*a**3*b**5*sin(c + d*x)**4*cos(c + d*x)**3 
/(3*d) - 224*a**3*b**5*sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - 64*a**3*b* 
*5*cos(c + d*x)**7/(15*d) + 35*a**2*b**6*x*sin(c + d*x)**8/32 + 35*a**2*b* 
*6*x*sin(c + d*x)**6*cos(c + d*x)**2/8 + 105*a**2*b**6*x*sin(c + d*x)**4*c 
os(c + d*x)**4/16 + 35*a**2*b**6*x*sin(c + d*x)**2*cos(c + d*x)**6/8 + 35* 
a**2*b**6*x*cos(c + d*x)**8/32 + 35*a**2*b**6*sin(c + d*x)**7*cos(c + d*x) 
/(32*d) - 511*a**2*b**6*sin(c + d*x)**5*cos(c + d*x)**3/(96*d) - 385*a**2* 
b**6*sin(c + d*x)**3*cos(c + d*x)**5/(96*d) - 35*a**2*b**6*sin(c + d*x)*co 
s(c + d*x)**7/(32*d) - 8*a*b**7*sin(c + d*x)**6*cos(c + d*x)**3/(3*d) - 16 
*a*b**7*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 64*a*b**7*sin(c + d*x)*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {1720320 \, a^{7} b \cos \left (d x + c\right )^{3} - 161280 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8} - 564480 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{6} b^{2} - 2408448 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{5} b^{3} + 235200 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{4} + 344064 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} b^{5} + 5880 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{6} - 16384 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a b^{7} - 21 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{8}}{645120 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

-1/645120*(1720320*a^7*b*cos(d*x + c)^3 - 161280*(2*d*x + 2*c + sin(2*d*x 
+ 2*c))*a^8 - 564480*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^6*b^2 - 2408448*(3 
*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^5*b^3 + 235200*(4*sin(2*d*x + 2*c)^3 
 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^4*b^4 + 344064*(15*cos(d*x + c)^7 
 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^3*b^5 + 5880*(64*sin(2*d*x + 2 
*c)^3 - 120*d*x - 120*c + 3*sin(8*d*x + 8*c) + 24*sin(4*d*x + 4*c))*a^2*b^ 
6 - 16384*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 1 
05*cos(d*x + c)^3)*a*b^7 - 21*(96*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c 
)^3 + 840*d*x + 840*c - 45*sin(8*d*x + 8*c) - 120*sin(4*d*x + 4*c))*b^8)/d
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {a b^{7} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} + \frac {b^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {1}{256} \, {\left (128 \, a^{8} + 896 \, a^{6} b^{2} + 1120 \, a^{4} b^{4} + 280 \, a^{2} b^{6} + 7 \, b^{8}\right )} x - \frac {{\left (28 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {{\left (28 \, a^{5} b^{3} + 21 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {{\left (16 \, a^{7} b + 28 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {{\left (32 \, a^{7} b + 112 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 7 \, a b^{7}\right )} \cos \left (d x + c\right )}{16 \, d} - \frac {{\left (56 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {{\left (1120 \, a^{4} b^{4} + 448 \, a^{2} b^{6} + 13 \, b^{8}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {{\left (224 \, a^{6} b^{2} + 280 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {{\left (128 \, a^{8} - 560 \, a^{4} b^{4} - 224 \, a^{2} b^{6} - 7 \, b^{8}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

1/288*a*b^7*cos(9*d*x + 9*c)/d + 1/5120*b^8*sin(10*d*x + 10*c)/d + 1/256*( 
128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x - 1/224*(28* 
a^3*b^5 + 5*a*b^7)*cos(7*d*x + 7*c)/d + 1/40*(28*a^5*b^3 + 21*a^3*b^5 + 2* 
a*b^7)*cos(5*d*x + 5*c)/d - 1/24*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*cos(3 
*d*x + 3*c)/d - 1/16*(32*a^7*b + 112*a^5*b^3 + 70*a^3*b^5 + 7*a*b^7)*cos(d 
*x + c)/d - 1/2048*(56*a^2*b^6 + 3*b^8)*sin(8*d*x + 8*c)/d + 1/3072*(1120* 
a^4*b^4 + 448*a^2*b^6 + 13*b^8)*sin(6*d*x + 6*c)/d - 1/256*(224*a^6*b^2 + 
280*a^4*b^4 + 56*a^2*b^6 + b^8)*sin(4*d*x + 4*c)/d + 1/512*(128*a^8 - 560* 
a^4*b^4 - 224*a^2*b^6 - 7*b^8)*sin(2*d*x + 2*c)/d
 

Mupad [B] (verification not implemented)

Time = 17.03 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {\frac {2205\,b^8\,\sin \left (2\,c+2\,d\,x\right )}{2}-20160\,a^8\,\sin \left (2\,c+2\,d\,x\right )+315\,b^8\,\sin \left (4\,c+4\,d\,x\right )-\frac {1365\,b^8\,\sin \left (6\,c+6\,d\,x\right )}{4}+\frac {945\,b^8\,\sin \left (8\,c+8\,d\,x\right )}{8}-\frac {63\,b^8\,\sin \left (10\,c+10\,d\,x\right )}{4}+53760\,a^7\,b\,\cos \left (3\,c+3\,d\,x\right )-4032\,a\,b^7\,\cos \left (5\,c+5\,d\,x\right )+1800\,a\,b^7\,\cos \left (7\,c+7\,d\,x\right )-280\,a\,b^7\,\cos \left (9\,c+9\,d\,x\right )+352800\,a^3\,b^5\,\cos \left (c+d\,x\right )+564480\,a^5\,b^3\,\cos \left (c+d\,x\right )+23520\,a^3\,b^5\,\cos \left (3\,c+3\,d\,x\right )+94080\,a^5\,b^3\,\cos \left (3\,c+3\,d\,x\right )-42336\,a^3\,b^5\,\cos \left (5\,c+5\,d\,x\right )-56448\,a^5\,b^3\,\cos \left (5\,c+5\,d\,x\right )+10080\,a^3\,b^5\,\cos \left (7\,c+7\,d\,x\right )+35280\,a^2\,b^6\,\sin \left (2\,c+2\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (2\,c+2\,d\,x\right )+17640\,a^2\,b^6\,\sin \left (4\,c+4\,d\,x\right )+88200\,a^4\,b^4\,\sin \left (4\,c+4\,d\,x\right )+70560\,a^6\,b^2\,\sin \left (4\,c+4\,d\,x\right )-11760\,a^2\,b^6\,\sin \left (6\,c+6\,d\,x\right )-29400\,a^4\,b^4\,\sin \left (6\,c+6\,d\,x\right )+2205\,a^2\,b^6\,\sin \left (8\,c+8\,d\,x\right )+35280\,a\,b^7\,\cos \left (c+d\,x\right )+161280\,a^7\,b\,\cos \left (c+d\,x\right )-40320\,a^8\,d\,x-2205\,b^8\,d\,x-88200\,a^2\,b^6\,d\,x-352800\,a^4\,b^4\,d\,x-282240\,a^6\,b^2\,d\,x}{80640\,d} \] Input:

int(cos(c + d*x)^2*(a + b*sin(c + d*x))^8,x)
 

Output:

-((2205*b^8*sin(2*c + 2*d*x))/2 - 20160*a^8*sin(2*c + 2*d*x) + 315*b^8*sin 
(4*c + 4*d*x) - (1365*b^8*sin(6*c + 6*d*x))/4 + (945*b^8*sin(8*c + 8*d*x)) 
/8 - (63*b^8*sin(10*c + 10*d*x))/4 + 53760*a^7*b*cos(3*c + 3*d*x) - 4032*a 
*b^7*cos(5*c + 5*d*x) + 1800*a*b^7*cos(7*c + 7*d*x) - 280*a*b^7*cos(9*c + 
9*d*x) + 352800*a^3*b^5*cos(c + d*x) + 564480*a^5*b^3*cos(c + d*x) + 23520 
*a^3*b^5*cos(3*c + 3*d*x) + 94080*a^5*b^3*cos(3*c + 3*d*x) - 42336*a^3*b^5 
*cos(5*c + 5*d*x) - 56448*a^5*b^3*cos(5*c + 5*d*x) + 10080*a^3*b^5*cos(7*c 
 + 7*d*x) + 35280*a^2*b^6*sin(2*c + 2*d*x) + 88200*a^4*b^4*sin(2*c + 2*d*x 
) + 17640*a^2*b^6*sin(4*c + 4*d*x) + 88200*a^4*b^4*sin(4*c + 4*d*x) + 7056 
0*a^6*b^2*sin(4*c + 4*d*x) - 11760*a^2*b^6*sin(6*c + 6*d*x) - 29400*a^4*b^ 
4*sin(6*c + 6*d*x) + 2205*a^2*b^6*sin(8*c + 8*d*x) + 35280*a*b^7*cos(c + d 
*x) + 161280*a^7*b*cos(c + d*x) - 40320*a^8*d*x - 2205*b^8*d*x - 88200*a^2 
*b^6*d*x - 352800*a^4*b^4*d*x - 282240*a^6*b^2*d*x)/(80640*d)
 

Reduce [B] (verification not implemented)

Time = 4.65 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.52 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

int(cos(d*x+c)^2*(a+b*sin(d*x+c))^8,x)
 

Output:

(8064*cos(c + d*x)*sin(c + d*x)**9*b**8 + 71680*cos(c + d*x)*sin(c + d*x)* 
*8*a*b**7 + 282240*cos(c + d*x)*sin(c + d*x)**7*a**2*b**6 - 1008*cos(c + d 
*x)*sin(c + d*x)**7*b**8 + 645120*cos(c + d*x)*sin(c + d*x)**6*a**3*b**5 - 
 10240*cos(c + d*x)*sin(c + d*x)**6*a*b**7 + 940800*cos(c + d*x)*sin(c + d 
*x)**5*a**4*b**4 - 47040*cos(c + d*x)*sin(c + d*x)**5*a**2*b**6 - 1176*cos 
(c + d*x)*sin(c + d*x)**5*b**8 + 903168*cos(c + d*x)*sin(c + d*x)**4*a**5* 
b**3 - 129024*cos(c + d*x)*sin(c + d*x)**4*a**3*b**5 - 12288*cos(c + d*x)* 
sin(c + d*x)**4*a*b**7 + 564480*cos(c + d*x)*sin(c + d*x)**3*a**6*b**2 - 2 
35200*cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 - 58800*cos(c + d*x)*sin(c + 
d*x)**3*a**2*b**6 - 1470*cos(c + d*x)*sin(c + d*x)**3*b**8 + 215040*cos(c 
+ d*x)*sin(c + d*x)**2*a**7*b - 301056*cos(c + d*x)*sin(c + d*x)**2*a**5*b 
**3 - 172032*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 16384*cos(c + d*x)*s 
in(c + d*x)**2*a*b**7 + 40320*cos(c + d*x)*sin(c + d*x)*a**8 - 282240*cos( 
c + d*x)*sin(c + d*x)*a**6*b**2 - 352800*cos(c + d*x)*sin(c + d*x)*a**4*b* 
*4 - 88200*cos(c + d*x)*sin(c + d*x)*a**2*b**6 - 2205*cos(c + d*x)*sin(c + 
 d*x)*b**8 - 215040*cos(c + d*x)*a**7*b - 602112*cos(c + d*x)*a**5*b**3 - 
344064*cos(c + d*x)*a**3*b**5 - 32768*cos(c + d*x)*a*b**7 + 40320*a**8*d*x 
 + 215040*a**7*b + 282240*a**6*b**2*d*x + 602112*a**5*b**3 + 352800*a**4*b 
**4*d*x + 344064*a**3*b**5 + 88200*a**2*b**6*d*x + 32768*a*b**7 + 2205*b** 
8*d*x)/(80640*d)