\(\int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx\) [42]

Optimal result

Integrand size = 21, antiderivative size = 286 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {4199 a^8 x}{1024}-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {4199 a^2 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^3}{6336 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d} \] Output:

4199/1024*a^8*x-4199/1920*a^8*cos(d*x+c)^5/d+4199/1024*a^8*cos(d*x+c)*sin( 
d*x+c)/d+4199/1536*a^8*cos(d*x+c)^3*sin(d*x+c)/d-323/1320*a^3*cos(d*x+c)^5 
*(a+a*sin(d*x+c))^5/d-19/132*a^2*cos(d*x+c)^5*(a+a*sin(d*x+c))^6/d-1/12*a* 
cos(d*x+c)^5*(a+a*sin(d*x+c))^7/d-4199/6336*a^2*cos(d*x+c)^5*(a^2+a^2*sin( 
d*x+c))^3/d-323/792*cos(d*x+c)^5*(a^2+a^2*sin(d*x+c))^4/d-4199/4032*cos(d* 
x+c)^5*(a^4+a^4*sin(d*x+c))^2/d-4199/2688*cos(d*x+c)^5*(a^8+a^8*sin(d*x+c) 
)/d
 

Mathematica [A] (verified)

Time = 3.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.74 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {a^8 \cos ^5(c+d x) \left (-29099070 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-22470656+11469281 \sin (c+d x)[phone] \sin ^2(c+d x)[phone] \sin ^3(c+d x)[phone] \sin ^4(c+d x)-8321928 \sin ^5(c+d x)-26346616 \sin ^6(c+d x)-20428112 \sin ^7(c+d x)-1239728 \sin ^8(c+d x)+9086336 \sin ^9(c+d x)+6969984 \sin ^{10}(c+d x)+2284800 \sin ^{11}(c+d x)+295680 \sin ^{12}(c+d x)\right )\right )}{3548160 d (-1+\sin (c+d x))^3 (1+\sin (c+d x))^{5/2}} \] Input:

Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]
 

Output:

-1/3548160*(a^8*Cos[c + d*x]^5*(-29099070*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sq 
rt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-22470656 + 114692 
81*Sin[c + d*x] + 13958687*Sin[c + d*x]^2 + 20459158*Sin[c + d*x]^3 + 1428 
3114*Sin[c + d*x]^4 - 8321928*Sin[c + d*x]^5 - 26346616*Sin[c + d*x]^6 - 2 
0428112*Sin[c + d*x]^7 - 1239728*Sin[c + d*x]^8 + 9086336*Sin[c + d*x]^9 + 
 6969984*Sin[c + d*x]^10 + 2284800*Sin[c + d*x]^11 + 295680*Sin[c + d*x]^1 
2)))/(d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.08, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3148, 3042, 3115, 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.

\(\displaystyle \int \cos ^4(c+d x) (a \sin (c+d x)+a)^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^4 (a \sin (c+d x)+a)^8dx\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^7dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^7dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^6dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^6dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^5dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^5dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^4dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^4dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^3dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^3dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)dx-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3148

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \int \cos ^4(c+d x)dx-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\)

Input:

Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]
 

Output:

-1/12*(a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^7)/d + (19*a*(-1/11*(a*Cos[c 
+ d*x]^5*(a + a*Sin[c + d*x])^6)/d + (17*a*(-1/10*(a*Cos[c + d*x]^5*(a + a 
*Sin[c + d*x])^5)/d + (3*a*(-1/9*(a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^4) 
/d + (13*a*(-1/8*(a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/d + (11*a*(-1/7 
*(a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/d + (9*a*(-1/6*(Cos[c + d*x]^5* 
(a^2 + a^2*Sin[c + d*x]))/d + (7*a*(-1/5*(a*Cos[c + d*x]^5)/d + a*((Cos[c 
+ d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d) 
))/4)))/6))/7))/8))/9))/2))/11))/12
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3157

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[a*((2*m + p - 1)/(m + p))   Int[(g* 
Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, 
g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers 
Q[2*m, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs. \(2(264)=528\).

Time = 0.07 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.87

\[\frac {a^{8} \left (-\frac {\sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{5}}{12}-\frac {7 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{5}}{120}-\frac {7 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{192}-\frac {7 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{384}+\frac {7 \left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{1536}+\frac {7 d x}{1024}+\frac {7 c}{1024}\right )+8 a^{8} \left (-\frac {\sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{5}}{11}-\frac {2 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{33}-\frac {8 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{231}-\frac {16 \cos \left (d x +c \right )^{5}}{1155}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{5}}{10}-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{16}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{32}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+56 a^{8} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )+70 a^{8} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+56 a^{8} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{7}-\frac {2 \cos \left (d x +c \right )^{5}}{35}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {8 a^{8} \cos \left (d x +c \right )^{5}}{5}+a^{8} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\]

Input:

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x)
 

Output:

1/d*(a^8*(-1/12*sin(d*x+c)^7*cos(d*x+c)^5-7/120*sin(d*x+c)^5*cos(d*x+c)^5- 
7/192*sin(d*x+c)^3*cos(d*x+c)^5-7/384*sin(d*x+c)*cos(d*x+c)^5+7/1536*(cos( 
d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+7/1024*d*x+7/1024*c)+8*a^8*(-1/11*sin( 
d*x+c)^6*cos(d*x+c)^5-2/33*sin(d*x+c)^4*cos(d*x+c)^5-8/231*sin(d*x+c)^2*co 
s(d*x+c)^5-16/1155*cos(d*x+c)^5)+28*a^8*(-1/10*sin(d*x+c)^5*cos(d*x+c)^5-1 
/16*sin(d*x+c)^3*cos(d*x+c)^5-1/32*sin(d*x+c)*cos(d*x+c)^5+1/128*(cos(d*x+ 
c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+56*a^8*(-1/9*sin(d*x+c) 
^4*cos(d*x+c)^5-4/63*sin(d*x+c)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+70*a^8* 
(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64*(cos(d*x 
+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+56*a^8*(-1/7*sin(d*x+c 
)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)+28*a^8*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1 
/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)-8/5*a^8*cos( 
d*x+c)^5+a^8*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.52 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {2580480 \, a^{8} \cos \left (d x + c\right )^{11} - 31539200 \, a^{8} \cos \left (d x + c\right )^{9} + 97320960 \, a^{8} \cos \left (d x + c\right )^{7} - 90832896 \, a^{8} \cos \left (d x + c\right )^{5} + 14549535 \, a^{8} d x + 231 \, {\left (1280 \, a^{8} \cos \left (d x + c\right )^{11} - 47744 \, a^{8} \cos \left (d x + c\right )^{9} + 253488 \, a^{8} \cos \left (d x + c\right )^{7} - 359624 \, a^{8} \cos \left (d x + c\right )^{5} + 41990 \, a^{8} \cos \left (d x + c\right )^{3} + 62985 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

1/3548160*(2580480*a^8*cos(d*x + c)^11 - 31539200*a^8*cos(d*x + c)^9 + 973 
20960*a^8*cos(d*x + c)^7 - 90832896*a^8*cos(d*x + c)^5 + 14549535*a^8*d*x 
+ 231*(1280*a^8*cos(d*x + c)^11 - 47744*a^8*cos(d*x + c)^9 + 253488*a^8*co 
s(d*x + c)^7 - 359624*a^8*cos(d*x + c)^5 + 41990*a^8*cos(d*x + c)^3 + 6298 
5*a^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. 1280 vs. \(2 (270) = 540\).

Time = 3.47 (sec) , antiderivative size = 1280, normalized size of antiderivative = 4.48 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c))**8,x)
 

Output:

Piecewise((7*a**8*x*sin(c + d*x)**12/1024 + 21*a**8*x*sin(c + d*x)**10*cos 
(c + d*x)**2/512 + 21*a**8*x*sin(c + d*x)**10/64 + 105*a**8*x*sin(c + d*x) 
**8*cos(c + d*x)**4/1024 + 105*a**8*x*sin(c + d*x)**8*cos(c + d*x)**2/64 + 
 105*a**8*x*sin(c + d*x)**8/64 + 35*a**8*x*sin(c + d*x)**6*cos(c + d*x)**6 
/256 + 105*a**8*x*sin(c + d*x)**6*cos(c + d*x)**4/32 + 105*a**8*x*sin(c + 
d*x)**6*cos(c + d*x)**2/16 + 7*a**8*x*sin(c + d*x)**6/4 + 105*a**8*x*sin(c 
 + d*x)**4*cos(c + d*x)**8/1024 + 105*a**8*x*sin(c + d*x)**4*cos(c + d*x)* 
*6/32 + 315*a**8*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 21*a**8*x*sin(c + 
d*x)**4*cos(c + d*x)**2/4 + 3*a**8*x*sin(c + d*x)**4/8 + 21*a**8*x*sin(c + 
 d*x)**2*cos(c + d*x)**10/512 + 105*a**8*x*sin(c + d*x)**2*cos(c + d*x)**8 
/64 + 105*a**8*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 21*a**8*x*sin(c + d* 
x)**2*cos(c + d*x)**4/4 + 3*a**8*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 7*a 
**8*x*cos(c + d*x)**12/1024 + 21*a**8*x*cos(c + d*x)**10/64 + 105*a**8*x*c 
os(c + d*x)**8/64 + 7*a**8*x*cos(c + d*x)**6/4 + 3*a**8*x*cos(c + d*x)**4/ 
8 + 7*a**8*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 119*a**8*sin(c + d*x)* 
*9*cos(c + d*x)**3/(3072*d) + 21*a**8*sin(c + d*x)**9*cos(c + d*x)/(64*d) 
- 281*a**8*sin(c + d*x)**7*cos(c + d*x)**5/(2560*d) + 49*a**8*sin(c + d*x) 
**7*cos(c + d*x)**3/(32*d) + 105*a**8*sin(c + d*x)**7*cos(c + d*x)/(64*d) 
- 8*a**8*sin(c + d*x)**6*cos(c + d*x)**5/(5*d) - 231*a**8*sin(c + d*x)**5* 
cos(c + d*x)**7/(2560*d) - 14*a**8*sin(c + d*x)**5*cos(c + d*x)**5/(5*d...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.19 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {45416448 \, a^{8} \cos \left (d x + c\right )^{5} - 196608 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 5046272 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} - 45416448 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 231 \, {\left (384 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 20 \, \sin \left (4 \, d x + 4 \, c\right )^{3} - 840 \, d x - 840 \, c - 15 \, \sin \left (8 \, d x + 8 \, c\right ) + 240 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 77616 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 4139520 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 1940400 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 887040 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{28385280 \, d} \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

-1/28385280*(45416448*a^8*cos(d*x + c)^5 - 196608*(105*cos(d*x + c)^11 - 3 
85*cos(d*x + c)^9 + 495*cos(d*x + c)^7 - 231*cos(d*x + c)^5)*a^8 + 5046272 
*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^8 - 4541644 
8*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^8 + 231*(384*sin(2*d*x + 2*c)^5 
+ 20*sin(4*d*x + 4*c)^3 - 840*d*x - 840*c - 15*sin(8*d*x + 8*c) + 240*sin( 
4*d*x + 4*c))*a^8 + 77616*(32*sin(2*d*x + 2*c)^5 - 120*d*x - 120*c - 5*sin 
(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*a^8 - 4139520*(4*sin(2*d*x + 2*c)^3 + 
 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^8 - 1940400*(24*d*x + 24*c + sin(8* 
d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^8 - 887040*(12*d*x + 12*c + sin(4*d*x + 
 4*c) + 8*sin(2*d*x + 2*c))*a^8)/d
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.73 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {4199}{1024} \, a^{8} x + \frac {a^{8} \cos \left (11 \, d x + 11 \, c\right )}{1408 \, d} - \frac {31 \, a^{8} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {139 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac {171 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{640 \, d} - \frac {323 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {323 \, a^{8} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{8} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {29 \, a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {673 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {361 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {8721 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {323 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

4199/1024*a^8*x + 1/1408*a^8*cos(11*d*x + 11*c)/d - 31/1152*a^8*cos(9*d*x 
+ 9*c)/d + 139/896*a^8*cos(7*d*x + 7*c)/d + 171/640*a^8*cos(5*d*x + 5*c)/d 
 - 323/192*a^8*cos(3*d*x + 3*c)/d - 323/64*a^8*cos(d*x + c)/d + 1/24576*a^ 
8*sin(12*d*x + 12*c)/d - 29/5120*a^8*sin(10*d*x + 10*c)/d + 673/8192*a^8*s 
in(8*d*x + 8*c)/d - 361/3072*a^8*sin(6*d*x + 6*c)/d - 8721/8192*a^8*sin(4* 
d*x + 4*c)/d + 323/512*a^8*sin(2*d*x + 2*c)/d
 

Mupad [B] (verification not implemented)

Time = 28.31 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.39 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^4*(a + a*sin(c + d*x))^8,x)
 

Output:

(4199*a^8*x)/1024 - ((1543*a^8*tan(c/2 + (d*x)/2)^3)/512 - (1068767*a^8*ta 
n(c/2 + (d*x)/2)^5)/2560 - (3297279*a^8*tan(c/2 + (d*x)/2)^7)/2560 - (1682 
83*a^8*tan(c/2 + (d*x)/2)^9)/3840 + (256139*a^8*tan(c/2 + (d*x)/2)^11)/256 
 - (256139*a^8*tan(c/2 + (d*x)/2)^13)/256 + (168283*a^8*tan(c/2 + (d*x)/2) 
^15)/3840 + (3297279*a^8*tan(c/2 + (d*x)/2)^17)/2560 + (1068767*a^8*tan(c/ 
2 + (d*x)/2)^19)/2560 - (1543*a^8*tan(c/2 + (d*x)/2)^21)/512 - (3175*a^8*t 
an(c/2 + (d*x)/2)^23)/512 + a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*(( 
4199*c)/1024 + (4199*d*x)/1024 - 43888/3465) + tan(c/2 + (d*x)/2)^22*(12*a 
^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((12597*c)/256 + (12597*d*x)/25 
6 - 16)) + tan(c/2 + (d*x)/2)^2*(12*a^8*((4199*c)/1024 + (4199*d*x)/1024) 
- a^8*((12597*c)/256 + (12597*d*x)/256 - 157072/1155)) + tan(c/2 + (d*x)/2 
)^20*(66*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((138567*c)/512 + (13 
8567*d*x)/512 - 336)) + tan(c/2 + (d*x)/2)^4*(66*a^8*((4199*c)/1024 + (419 
9*d*x)/1024) - a^8*((138567*c)/512 + (138567*d*x)/512 - 52496/105)) + tan( 
c/2 + (d*x)/2)^18*(220*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((23094 
5*c)/256 + (230945*d*x)/256 - 5584/3)) + tan(c/2 + (d*x)/2)^6*(220*a^8*((4 
199*c)/1024 + (4199*d*x)/1024) - a^8*((230945*c)/256 + (230945*d*x)/256 - 
58288/63)) + tan(c/2 + (d*x)/2)^14*(792*a^8*((4199*c)/1024 + (4199*d*x)/10 
24) - a^8*((415701*c)/128 + (415701*d*x)/128 - 17696/5)) + tan(c/2 + (d*x) 
/2)^10*(792*a^8*((4199*c)/1024 + (4199*d*x)/1024) - a^8*((415701*c)/128...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.69 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^{8} \left (-295680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11}-2580480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10}-9550464 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-18636800 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-17397072 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+3031040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}[phone] \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}[phone] \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}[phone] \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+2957312 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-11001375 \cos \left (d x +c \right ) \sin \left (d x +c \right )-22470656 \cos \left (d x +c \right )[phone] d x [phone]\right )}{3548160 d} \] Input:

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^8,x)
 

Output:

(a**8*( - 295680*cos(c + d*x)*sin(c + d*x)**11 - 2580480*cos(c + d*x)*sin( 
c + d*x)**10 - 9550464*cos(c + d*x)*sin(c + d*x)**9 - 18636800*cos(c + d*x 
)*sin(c + d*x)**8 - 17397072*cos(c + d*x)*sin(c + d*x)**7 + 3031040*cos(c 
+ d*x)*sin(c + d*x)**6 + 29377656*cos(c + d*x)*sin(c + d*x)**5 + 37699584* 
cos(c + d*x)*sin(c + d*x)**4 + 23416470*cos(c + d*x)*sin(c + d*x)**3 + 295 
7312*cos(c + d*x)*sin(c + d*x)**2 - 11001375*cos(c + d*x)*sin(c + d*x) - 2 
2470656*cos(c + d*x) + 14549535*d*x + 22470656))/(3548160*d)