Integrand size = 21, antiderivative size = 286 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {4199 a^8 x}{1024}-\frac {4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac {4199 a^8 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {4199 a^8 \cos ^3(c+d x) \sin (c+d x)}{1536 d}-\frac {323 a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^5}{1320 d}-\frac {19 a^2 \cos ^5(c+d x) (a+a \sin (c+d x))^6}{132 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^7}{12 d}-\frac {4199 a^2 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^3}{6336 d}-\frac {323 \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{792 d}-\frac {4199 \cos ^5(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{4032 d}-\frac {4199 \cos ^5(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{2688 d} \]
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
Time = 2.00 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.74 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {a^8 \cos ^5(c+d x) \left (-29099070 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-22470656+11469281 \sin (c+d x)+13958687 \sin ^2(c+d x)+20459158 \sin ^3(c+d x)+14283114 \sin ^4(c+d x)-8321928 \sin ^5(c+d x)-26346616 \sin ^6(c+d x)-20428112 \sin ^7(c+d x)-1239728 \sin ^8(c+d x)+9086336 \sin ^9(c+d x)+6969984 \sin ^{10}(c+d x)+2284800 \sin ^{11}(c+d x)+295680 \sin ^{12}(c+d x)\right )\right )}{3548160 d (-1+\sin (c+d x))^3 (1+\sin (c+d x))^{5/2}} \]
-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))
Time = 1.57 (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 |
| \(\displaystyle \frac {19}{12} a \left (\frac {17}{11} a \left (\frac {3}{2} a \left (\frac {13}{9} a \left (\frac {11}{8} a \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^4}{9 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{11 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}\) |
-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
3.1.42.3.1 Defintions of rubi rules used
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]
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])
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]
Time = 2.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.50
| method | result | size |
| parallelrisch | \(\frac {323 \left (\frac {13 d x}{2}+\sin \left (2 d x +2 c \right )-\frac {27 \sin \left (4 d x +4 c \right )}{16}-\frac {19 \sin \left (6 d x +6 c \right )}{102}+\frac {673 \sin \left (8 d x +8 c \right )}{5168}+\frac {4 \cos \left (11 d x +11 c \right )}{3553}-\frac {29 \sin \left (10 d x +10 c \right )}{3230}+\frac {\sin \left (12 d x +12 c \right )}{15504}-8 \cos \left (d x +c \right )-\frac {8 \cos \left (3 d x +3 c \right )}{3}+\frac {36 \cos \left (5 d x +5 c \right )}{85}+\frac {556 \cos \left (7 d x +7 c \right )}{2261}-\frac {124 \cos \left (9 d x +9 c \right )}{2907}-\frac {11235328}{1119195}\right ) a^{8}}{512 d}\) | \(142\) |
| risch | \(-\frac {29 a^{8} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {323 a^{8} \cos \left (d x +c \right )}{64 d}+\frac {4199 a^{8} x}{1024}+\frac {a^{8} \cos \left (11 d x +11 c \right )}{1408 d}+\frac {a^{8} \sin \left (12 d x +12 c \right )}{24576 d}-\frac {31 a^{8} \cos \left (9 d x +9 c \right )}{1152 d}+\frac {673 a^{8} \sin \left (8 d x +8 c \right )}{8192 d}+\frac {139 a^{8} \cos \left (7 d x +7 c \right )}{896 d}-\frac {361 a^{8} \sin \left (6 d x +6 c \right )}{3072 d}+\frac {171 a^{8} \cos \left (5 d x +5 c \right )}{640 d}-\frac {8721 a^{8} \sin \left (4 d x +4 c \right )}{8192 d}-\frac {323 a^{8} \cos \left (3 d x +3 c \right )}{192 d}+\frac {323 a^{8} \sin \left (2 d x +2 c \right )}{512 d}\) | \(209\) |
| derivativedivides | \(\frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{12}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{120}-\frac {7 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{192}-\frac {7 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{384}+\frac {7 \left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+a^{8} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\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}\) | \(535\) |
| default | \(\frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{12}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{120}-\frac {7 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{192}-\frac {7 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{384}+\frac {7 \left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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 {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\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} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+a^{8} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\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}\) | \(535\) |
323/512*(13/2*d*x+sin(2*d*x+2*c)-27/16*sin(4*d*x+4*c)-19/102*sin(6*d*x+6*c )+673/5168*sin(8*d*x+8*c)+4/3553*cos(11*d*x+11*c)-29/3230*sin(10*d*x+10*c) +1/15504*sin(12*d*x+12*c)-8*cos(d*x+c)-8/3*cos(3*d*x+3*c)+36/85*cos(5*d*x+ 5*c)+556/2261*cos(7*d*x+7*c)-124/2907*cos(9*d*x+9*c)-11235328/1119195)*a^8 /d
Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.52 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {2580480 \, a^{8} \cos \left (d x + c\right )^{11} - 31539200 \, a^{8} \cos \left (d x + c\right )^{9} + 97320960 \, a^{8} \cos \left (d x + c\right )^{7} - 90832896 \, a^{8} \cos \left (d x + c\right )^{5} + 14549535 \, a^{8} d x + 231 \, {\left (1280 \, a^{8} \cos \left (d x + c\right )^{11} - 47744 \, a^{8} \cos \left (d x + c\right )^{9} + 253488 \, a^{8} \cos \left (d x + c\right )^{7} - 359624 \, a^{8} \cos \left (d x + c\right )^{5} + 41990 \, a^{8} \cos \left (d x + c\right )^{3} + 62985 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (270) = 540\).
Time = 2.81 (sec) , antiderivative size = 1280, normalized size of antiderivative = 4.48 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Too large to display} \]
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...
Time = 0.21 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.19 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {45416448 \, a^{8} \cos \left (d x + c\right )^{5} - 196608 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 5046272 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} - 45416448 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 231 \, {\left (384 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 20 \, \sin \left (4 \, d x + 4 \, c\right )^{3} - 840 \, d x - 840 \, c - 15 \, \sin \left (8 \, d x + 8 \, c\right ) + 240 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 77616 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 4139520 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 1940400 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 887040 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{28385280 \, d} \]
-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
Time = 0.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.73 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {4199}{1024} \, a^{8} x + \frac {a^{8} \cos \left (11 \, d x + 11 \, c\right )}{1408 \, d} - \frac {31 \, a^{8} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {139 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac {171 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{640 \, d} - \frac {323 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {323 \, a^{8} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{8} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {29 \, a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {673 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {361 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {8721 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {323 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
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
Time = 9.07 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.39 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Too large to display} \]
(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...