\(\int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx\) [44]

Optimal result

Integrand size = 21, antiderivative size = 262 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {2431 a^8 x}{256}-\frac {2431 a^8 \cos ^3(c+d x)}{384 d}+\frac {2431 a^8 \cos (c+d x) \sin (c+d x)}{256 d}-\frac {17 a^3 \cos ^3(c+d x) (a+a \sin (c+d x))^5}{48 d}-\frac {17 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^6}{90 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^7}{10 d}-\frac {2431 a^2 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^3}{2016 d}-\frac {221 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^4}{336 d}-\frac {2431 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )^2}{1120 d}-\frac {2431 \cos ^3(c+d x) \left (a^8+a^8 \sin (c+d x)\right )}{640 d} \] Output:

2431/256*a^8*x-2431/384*a^8*cos(d*x+c)^3/d+2431/256*a^8*cos(d*x+c)*sin(d*x 
+c)/d-17/48*a^3*cos(d*x+c)^3*(a+a*sin(d*x+c))^5/d-17/90*a^2*cos(d*x+c)^3*( 
a+a*sin(d*x+c))^6/d-1/10*a*cos(d*x+c)^3*(a+a*sin(d*x+c))^7/d-2431/2016*a^2 
*cos(d*x+c)^3*(a^2+a^2*sin(d*x+c))^3/d-221/336*cos(d*x+c)^3*(a^2+a^2*sin(d 
*x+c))^4/d-2431/1120*cos(d*x+c)^3*(a^4+a^4*sin(d*x+c))^2/d-2431/640*cos(d* 
x+c)^3*(a^8+a^8*sin(d*x+c))/d
 

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {a^8 \cos ^3(c+d x) \left (1531530 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (1193984-508859 \sin (c+d x)-410693 \sin ^2(c+d x)-543442 \sin ^3(c+d x)-492846 \sin ^4(c+d x)-130728 \sin ^5(c+d x)+257704 \sin ^6(c+d x)+353648 \sin ^7(c+d x)+209552 \sin ^8(c+d x)+63616 \sin ^9(c+d x)+8064 \sin ^{10}(c+d x)\right )\right )}{80640 d (-1+\sin (c+d x))^2 (1+\sin (c+d x))^{3/2}} \] Input:

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

Output:

-1/80640*(a^8*Cos[c + d*x]^3*(1531530*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2 
]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(1193984 - 508859*Sin[c 
 + d*x] - 410693*Sin[c + d*x]^2 - 543442*Sin[c + d*x]^3 - 492846*Sin[c + d 
*x]^4 - 130728*Sin[c + d*x]^5 + 257704*Sin[c + d*x]^6 + 353648*Sin[c + d*x 
]^7 + 209552*Sin[c + d*x]^8 + 63616*Sin[c + d*x]^9 + 8064*Sin[c + d*x]^10) 
))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 3042, 3157, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3148

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

Input:

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

Output:

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

Time = 0.06 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.83

\[\frac {a^{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 )+8 a^{8} \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 )+28 a^{8} \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 )+56 a^{8} \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 )+70 a^{8} \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^{8} \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 )+28 a^{8} \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 )-\frac {8 a^{8} \cos \left (d x +c \right )^{3}}{3}+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 )}{d}\]

Input:

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

Output:

1/d*(a^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)+8*a^8*(-1/9*sin(d*x+c)^6*cos(d*x+c)^3-2/2 
1*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)+28*a^8*(-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)*cos(d*x+c)^3+5/128*cos(d*x+c)*sin(d*x+c)+5/128*d*x+5/12 
8*c)+56*a^8*(-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)+70*a^8*(-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*cos(d*x+c)*sin(d*x+c)+1/16*d*x+1/16*c)+56*a^8*(-1/5*sin 
(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+28*a^8*(-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)-8/3*a^8*cos(d*x+c)^3+a^8*(1/ 
2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.52 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {71680 \, a^{8} \cos \left (d x + c\right )^{9} - 921600 \, a^{8} \cos \left (d x + c\right )^{7} + 3096576 \, a^{8} \cos \left (d x + c\right )^{5} - 3440640 \, a^{8} \cos \left (d x + c\right )^{3} + 765765 \, a^{8} d x + 63 \, {\left (128 \, a^{8} \cos \left (d x + c\right )^{9} - 4976 \, a^{8} \cos \left (d x + c\right )^{7} + 28328 \, a^{8} \cos \left (d x + c\right )^{5} - 46510 \, a^{8} \cos \left (d x + c\right )^{3} + 12155 \, a^{8} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \] Input:

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

Output:

1/80640*(71680*a^8*cos(d*x + c)^9 - 921600*a^8*cos(d*x + c)^7 + 3096576*a^ 
8*cos(d*x + c)^5 - 3440640*a^8*cos(d*x + c)^3 + 765765*a^8*d*x + 63*(128*a 
^8*cos(d*x + c)^9 - 4976*a^8*cos(d*x + c)^7 + 28328*a^8*cos(d*x + c)^5 - 4 
6510*a^8*cos(d*x + c)^3 + 12155*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. 1018 vs. \(2 (246) = 492\).

Time = 1.97 (sec) , antiderivative size = 1018, normalized size of antiderivative = 3.89 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((7*a**8*x*sin(c + d*x)**10/256 + 35*a**8*x*sin(c + d*x)**8*cos(c 
 + d*x)**2/256 + 35*a**8*x*sin(c + d*x)**8/32 + 35*a**8*x*sin(c + d*x)**6* 
cos(c + d*x)**4/128 + 35*a**8*x*sin(c + d*x)**6*cos(c + d*x)**2/8 + 35*a** 
8*x*sin(c + d*x)**6/8 + 35*a**8*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 10 
5*a**8*x*sin(c + d*x)**4*cos(c + d*x)**4/16 + 105*a**8*x*sin(c + d*x)**4*c 
os(c + d*x)**2/8 + 7*a**8*x*sin(c + d*x)**4/2 + 35*a**8*x*sin(c + d*x)**2* 
cos(c + d*x)**8/256 + 35*a**8*x*sin(c + d*x)**2*cos(c + d*x)**6/8 + 105*a* 
*8*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + 7*a**8*x*sin(c + d*x)**2*cos(c + 
d*x)**2 + a**8*x*sin(c + d*x)**2/2 + 7*a**8*x*cos(c + d*x)**10/256 + 35*a* 
*8*x*cos(c + d*x)**8/32 + 35*a**8*x*cos(c + d*x)**6/8 + 7*a**8*x*cos(c + d 
*x)**4/2 + a**8*x*cos(c + d*x)**2/2 + 7*a**8*sin(c + d*x)**9*cos(c + d*x)/ 
(256*d) - 79*a**8*sin(c + d*x)**7*cos(c + d*x)**3/(384*d) + 35*a**8*sin(c 
+ d*x)**7*cos(c + d*x)/(32*d) - 8*a**8*sin(c + d*x)**6*cos(c + d*x)**3/(3* 
d) - 7*a**8*sin(c + d*x)**5*cos(c + d*x)**5/(30*d) - 511*a**8*sin(c + d*x) 
**5*cos(c + d*x)**3/(96*d) + 35*a**8*sin(c + d*x)**5*cos(c + d*x)/(8*d) - 
16*a**8*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 56*a**8*sin(c + d*x)**4*co 
s(c + d*x)**3/(3*d) - 49*a**8*sin(c + d*x)**3*cos(c + d*x)**7/(384*d) - 38 
5*a**8*sin(c + d*x)**3*cos(c + d*x)**5/(96*d) - 35*a**8*sin(c + d*x)**3*co 
s(c + d*x)**3/(3*d) + 7*a**8*sin(c + d*x)**3*cos(c + d*x)/(2*d) - 64*a**8* 
sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 224*a**8*sin(c + d*x)**2*cos(c...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.22 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=-\frac {1720320 \, a^{8} \cos \left (d x + c\right )^{3} - 16384 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 344064 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 2408448 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{8} - 21 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c - 45 \, \sin \left (8 \, d x + 8 \, c\right ) - 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 5880 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 120 \, d x - 120 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} + 235200 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 564480 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{8} - 161280 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{8}}{645120 \, d} \] Input:

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

Output:

-1/645120*(1720320*a^8*cos(d*x + c)^3 - 16384*(35*cos(d*x + c)^9 - 135*cos 
(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a^8 + 344064*(15*co 
s(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^8 - 2408448*(3*cos 
(d*x + c)^5 - 5*cos(d*x + c)^3)*a^8 - 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))*a^8 + 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^8 + 235200*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 1 
2*c + 3*sin(4*d*x + 4*c))*a^8 - 564480*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^ 
8 - 161280*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^8)/d
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.66 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {2431}{256} \, a^{8} x + \frac {a^{8} \cos \left (9 \, d x + 9 \, c\right )}{288 \, d} - \frac {33 \, a^{8} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {51 \, a^{8} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {17 \, a^{8} \cos \left (3 \, d x + 3 \, c\right )}{8 \, d} - \frac {221 \, a^{8} \cos \left (d x + c\right )}{16 \, d} + \frac {a^{8} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {59 \, a^{8} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {527 \, a^{8} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {561 \, a^{8} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {663 \, a^{8} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \] Input:

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

Output:

2431/256*a^8*x + 1/288*a^8*cos(9*d*x + 9*c)/d - 33/224*a^8*cos(7*d*x + 7*c 
)/d + 51/40*a^8*cos(5*d*x + 5*c)/d - 17/8*a^8*cos(3*d*x + 3*c)/d - 221/16* 
a^8*cos(d*x + c)/d + 1/5120*a^8*sin(10*d*x + 10*c)/d - 59/2048*a^8*sin(8*d 
*x + 8*c)/d + 527/1024*a^8*sin(6*d*x + 6*c)/d - 561/256*a^8*sin(4*d*x + 4* 
c)/d - 663/512*a^8*sin(2*d*x + 2*c)/d
 

Mupad [B] (verification not implemented)

Time = 28.18 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.18 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:

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

Output:

(2431*a^8*x)/256 - ((11809*a^8*tan(c/2 + (d*x)/2)^3)/128 - (23647*a^8*tan( 
c/2 + (d*x)/2)^5)/160 - (40749*a^8*tan(c/2 + (d*x)/2)^7)/32 - (70499*a^8*t 
an(c/2 + (d*x)/2)^9)/64 + (70499*a^8*tan(c/2 + (d*x)/2)^11)/64 + (40749*a^ 
8*tan(c/2 + (d*x)/2)^13)/32 + (23647*a^8*tan(c/2 + (d*x)/2)^15)/160 - (118 
09*a^8*tan(c/2 + (d*x)/2)^17)/128 - (2175*a^8*tan(c/2 + (d*x)/2)^19)/128 + 
 a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((2431*c)/256 + (2431*d*x)/256 
- 9328/315) + tan(c/2 + (d*x)/2)^18*(10*a^8*((2431*c)/256 + (2431*d*x)/256 
) - a^8*((12155*c)/128 + (12155*d*x)/128 - 16)) + tan(c/2 + (d*x)/2)^2*(10 
*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((12155*c)/128 + (12155*d*x)/12 
8 - 17648/63)) + tan(c/2 + (d*x)/2)^14*(120*a^8*((2431*c)/256 + (2431*d*x) 
/256) - a^8*((36465*c)/32 + (36465*d*x)/32 - 1984)) + tan(c/2 + (d*x)/2)^6 
*(120*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((36465*c)/32 + (36465*d*x 
)/32 - 32960/21)) + tan(c/2 + (d*x)/2)^16*(45*a^8*((2431*c)/256 + (2431*d* 
x)/256) - a^8*((109395*c)/256 + (109395*d*x)/256 - 336)) + tan(c/2 + (d*x) 
/2)^4*(45*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((109395*c)/256 + (109 
395*d*x)/256 - 6976/7)) + tan(c/2 + (d*x)/2)^10*(252*a^8*((2431*c)/256 + ( 
2431*d*x)/256) - a^8*((153153*c)/64 + (153153*d*x)/64 - 18656/5)) + tan(c/ 
2 + (d*x)/2)^12*(210*a^8*((2431*c)/256 + (2431*d*x)/256) - a^8*((255255*c) 
/128 + (255255*d*x)/128 - 4288)) + tan(c/2 + (d*x)/2)^8*(210*a^8*((2431*c) 
/256 + (2431*d*x)/256) - a^8*((255255*c)/128 + (255255*d*x)/128 - 5792/...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx=\frac {a^{8} \left (8064 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}+71680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+281232 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+634880 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+892584 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+761856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+269010 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-274432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-685125 \cos \left (d x +c \right ) \sin \left (d x +c \right )-1193984 \cos \left (d x +c \right )+765765 d x +1193984\right )}{80640 d} \] Input:

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

Output:

(a**8*(8064*cos(c + d*x)*sin(c + d*x)**9 + 71680*cos(c + d*x)*sin(c + d*x) 
**8 + 281232*cos(c + d*x)*sin(c + d*x)**7 + 634880*cos(c + d*x)*sin(c + d* 
x)**6 + 892584*cos(c + d*x)*sin(c + d*x)**5 + 761856*cos(c + d*x)*sin(c + 
d*x)**4 + 269010*cos(c + d*x)*sin(c + d*x)**3 - 274432*cos(c + d*x)*sin(c 
+ d*x)**2 - 685125*cos(c + d*x)*sin(c + d*x) - 1193984*cos(c + d*x) + 7657 
65*d*x + 1193984))/(80640*d)