\(\int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1016]

Optimal result

Integrand size = 31, antiderivative size = 159 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {8 (A-B) (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}-\frac {4 (3 A-5 B) (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}+\frac {6 (A-3 B) (a+a \sin (e+f x))^{6+m}}{a^6 f (6+m)}-\frac {(A-7 B) (a+a \sin (e+f x))^{7+m}}{a^7 f (7+m)}-\frac {B (a+a \sin (e+f x))^{8+m}}{a^8 f (8+m)} \] Output:

8*(A-B)*(a+a*sin(f*x+e))^(4+m)/a^4/f/(4+m)-4*(3*A-5*B)*(a+a*sin(f*x+e))^(5 
+m)/a^5/f/(5+m)+6*(A-3*B)*(a+a*sin(f*x+e))^(6+m)/a^6/f/(6+m)-(A-7*B)*(a+a* 
sin(f*x+e))^(7+m)/a^7/f/(7+m)-B*(a+a*sin(f*x+e))^(8+m)/a^8/f/(8+m)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.83 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {(a (1+\sin (e+f x)))^{4+m} \left (\frac {8 a^4 (A-B)}{4+m}-\frac {4 a^4 (3 A-5 B) (1+\sin (e+f x))}{5+m}+\frac {6 a^4 (A-3 B) (1+\sin (e+f x))^2}{6+m}-\frac {a^4 (A-7 B) (1+\sin (e+f x))^3}{7+m}-\frac {B (a+a \sin (e+f x))^4}{8+m}\right )}{a^8 f} \] Input:

Integrate[Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]
 

Output:

((a*(1 + Sin[e + f*x]))^(4 + m)*((8*a^4*(A - B))/(4 + m) - (4*a^4*(3*A - 5 
*B)*(1 + Sin[e + f*x]))/(5 + m) + (6*a^4*(A - 3*B)*(1 + Sin[e + f*x])^2)/( 
6 + m) - (a^4*(A - 7*B)*(1 + Sin[e + f*x])^3)/(7 + m) - (B*(a + a*Sin[e + 
f*x])^4)/(8 + m)))/(a^8*f)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 3315, 27, 86, 2009}

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.

\(\displaystyle \int \cos ^7(e+f x) (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (e+f x)^7 (a \sin (e+f x)+a)^m (A+B \sin (e+f x))dx\)

\(\Big \downarrow \) 3315

\(\displaystyle \frac {\int \frac {(a-a \sin (e+f x))^3 (\sin (e+f x) a+a)^{m+3} (a A+a B \sin (e+f x))}{a}d(a \sin (e+f x))}{a^7 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (a-a \sin (e+f x))^3 (\sin (e+f x) a+a)^{m+3} (a A+a B \sin (e+f x))d(a \sin (e+f x))}{a^8 f}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\int \left (8 a^4 (A-B) (\sin (e+f x) a+a)^{m+3}-4 a^3 (3 A-5 B) (\sin (e+f x) a+a)^{m+4}+6 a^2 (A-3 B) (\sin (e+f x) a+a)^{m+5}-a (A-7 B) (\sin (e+f x) a+a)^{m+6}-B (\sin (e+f x) a+a)^{m+7}\right )d(a \sin (e+f x))}{a^8 f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {8 a^4 (A-B) (a \sin (e+f x)+a)^{m+4}}{m+4}-\frac {4 a^3 (3 A-5 B) (a \sin (e+f x)+a)^{m+5}}{m+5}+\frac {6 a^2 (A-3 B) (a \sin (e+f x)+a)^{m+6}}{m+6}-\frac {a (A-7 B) (a \sin (e+f x)+a)^{m+7}}{m+7}-\frac {B (a \sin (e+f x)+a)^{m+8}}{m+8}}{a^8 f}\)

Input:

Int[Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]
 

Output:

((8*a^4*(A - B)*(a + a*Sin[e + f*x])^(4 + m))/(4 + m) - (4*a^3*(3*A - 5*B) 
*(a + a*Sin[e + f*x])^(5 + m))/(5 + m) + (6*a^2*(A - 3*B)*(a + a*Sin[e + f 
*x])^(6 + m))/(6 + m) - (a*(A - 7*B)*(a + a*Sin[e + f*x])^(7 + m))/(7 + m) 
 - (B*(a + a*Sin[e + f*x])^(8 + m))/(8 + m))/(a^8*f)
 

Defintions of rubi rules used

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3315

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* 
f)   Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, 
 x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege 
rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs. \(2(159)=318\).

Time = 21.58 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.43

method	result	size
parallelrisch	\(\frac {\left (\left (\left (15 A +B \right ) m^{4}+\left (447 A +19 B \right ) m^{3}+\left (5028 A -94 B \right ) m^{2}+\left (19296 A -8932 B \right ) m -11760 B \right ) \cos \left (2 f x +2 e \right )+\left (\left (6 A -B \right ) m^{4}+\left (150 A -52 B \right ) m^{3}+\left (1080 A -989 B \right ) m^{2}+\left (2112 A -4466 B \right ) m -5880 B \right ) \cos \left (4 f x +4 e \right )+\left (4+m \right ) \left (5+m \right ) \left (\left (A -B \right ) m^{2}+\left (8 A -26 B \right ) m -84 B \right ) \cos \left (6 f x +6 e \right )+\frac {9 \left (\left (A +B \right ) m +8 A \right ) \left (m^{3}+31 m^{2}+\frac {1070}{3} m +\frac {1960}{3}\right ) \sin \left (3 f x +3 e \right )}{2}+\frac {5 \left (4+m \right ) \left (\left (A +B \right ) m +8 A \right ) \left (m^{2}+\frac {103}{5} m +\frac {294}{5}\right ) \sin \left (5 f x +5 e \right )}{2}+\frac {\left (4+m \right ) \left (6+m \right ) \left (\left (A +B \right ) m +8 A \right ) \left (5+m \right ) \sin \left (7 f x +7 e \right )}{2}-\frac {B \left (5+m \right ) \left (4+m \right ) \left (7+m \right ) \left (6+m \right ) \cos \left (8 f x +8 e \right )}{4}+\frac {5 \left (\left (A +B \right ) m +8 A \right ) \left (m^{3}+\frac {171}{5} m^{2}+\frac {2578}{5} m +5880\right ) \sin \left (f x +e \right )}{2}+\left (\frac {5 B}{4}+10 A \right ) m^{4}+\left (\frac {83 B}{2}+314 A \right ) m^{3}+\left (4040 A +\frac {2407 B}{4}\right ) m^{2}+\left (29632 A +\frac {13411 B}{2}\right ) m +98304 A -7350 B \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m}}{32 \left (5+m \right ) \left (4+m \right ) \left (8+m \right ) \left (7+m \right ) \left (6+m \right ) f}\)	\(387\)
derivativedivides	\(\frac {\left (A \,m^{4}+29 A \,m^{3}-B \,m^{3}+320 A \,m^{2}-27 B \,m^{2}+1600 A m -254 B m +3072 A -840 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}+\frac {\left (A \,m^{4}+B \,m^{4}+35 A \,m^{3}+27 B \,m^{3}+470 A \,m^{2}+254 B \,m^{2}+2872 A m +840 B m +6720 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {B \sin \left (f x +e \right )^{8} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (8+m \right )}+\frac {3 \left (A \,m^{4}-B \,m^{4}+21 A \,m^{3}-31 B \,m^{3}+136 A \,m^{2}-326 B \,m^{2}+256 A m -1276 B m -1680 B \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {\left (3 A \,m^{4}-B \,m^{4}+75 A \,m^{3}-37 B \,m^{3}+636 A \,m^{2}-488 B \,m^{2}+1824 A m -2552 B m -3360 B \right ) \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {3 \left (A \,m^{4}+B \,m^{4}+31 A \,m^{3}+23 B \,m^{3}+346 A \,m^{2}+162 B \,m^{2}+1576 A m +280 B m +2240 A \right ) \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {\left (A m +B m +8 A \right ) \sin \left (f x +e \right )^{7} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+15 m +56\right )}-\frac {\left (A \,m^{2}-3 B \,m^{2}+8 A m -52 B m -168 B \right ) \sin \left (f x +e \right )^{6} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+21 m^{2}+146 m +336\right )}+\frac {3 \left (A \,m^{3}+B \,m^{3}+23 A \,m^{2}+15 B \,m^{2}+162 A m +42 B m +336 A \right ) \sin \left (f x +e \right )^{5} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+26 m^{3}+251 m^{2}+1066 m +1680\right )}\)	\(707\)
default	\(\frac {\left (A \,m^{4}+29 A \,m^{3}-B \,m^{3}+320 A \,m^{2}-27 B \,m^{2}+1600 A m -254 B m +3072 A -840 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}+\frac {\left (A \,m^{4}+B \,m^{4}+35 A \,m^{3}+27 B \,m^{3}+470 A \,m^{2}+254 B \,m^{2}+2872 A m +840 B m +6720 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {B \sin \left (f x +e \right )^{8} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (8+m \right )}+\frac {3 \left (A \,m^{4}-B \,m^{4}+21 A \,m^{3}-31 B \,m^{3}+136 A \,m^{2}-326 B \,m^{2}+256 A m -1276 B m -1680 B \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {\left (3 A \,m^{4}-B \,m^{4}+75 A \,m^{3}-37 B \,m^{3}+636 A \,m^{2}-488 B \,m^{2}+1824 A m -2552 B m -3360 B \right ) \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {3 \left (A \,m^{4}+B \,m^{4}+31 A \,m^{3}+23 B \,m^{3}+346 A \,m^{2}+162 B \,m^{2}+1576 A m +280 B m +2240 A \right ) \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720\right )}-\frac {\left (A m +B m +8 A \right ) \sin \left (f x +e \right )^{7} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+15 m +56\right )}-\frac {\left (A \,m^{2}-3 B \,m^{2}+8 A m -52 B m -168 B \right ) \sin \left (f x +e \right )^{6} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+21 m^{2}+146 m +336\right )}+\frac {3 \left (A \,m^{3}+B \,m^{3}+23 A \,m^{2}+15 B \,m^{2}+162 A m +42 B m +336 A \right ) \sin \left (f x +e \right )^{5} {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+26 m^{3}+251 m^{2}+1066 m +1680\right )}\)	\(707\)

Input:

int(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x,method=_RETURNVERBO 
SE)
 

Output:

1/32*(((15*A+B)*m^4+(447*A+19*B)*m^3+(5028*A-94*B)*m^2+(19296*A-8932*B)*m- 
11760*B)*cos(2*f*x+2*e)+((6*A-B)*m^4+(150*A-52*B)*m^3+(1080*A-989*B)*m^2+( 
2112*A-4466*B)*m-5880*B)*cos(4*f*x+4*e)+(4+m)*(5+m)*((A-B)*m^2+(8*A-26*B)* 
m-84*B)*cos(6*f*x+6*e)+9/2*((A+B)*m+8*A)*(m^3+31*m^2+1070/3*m+1960/3)*sin( 
3*f*x+3*e)+5/2*(4+m)*((A+B)*m+8*A)*(m^2+103/5*m+294/5)*sin(5*f*x+5*e)+1/2* 
(4+m)*(6+m)*((A+B)*m+8*A)*(5+m)*sin(7*f*x+7*e)-1/4*B*(5+m)*(4+m)*(7+m)*(6+ 
m)*cos(8*f*x+8*e)+5/2*((A+B)*m+8*A)*(m^3+171/5*m^2+2578/5*m+5880)*sin(f*x+ 
e)+(5/4*B+10*A)*m^4+(83/2*B+314*A)*m^3+(4040*A+2407/4*B)*m^2+(29632*A+1341 
1/2*B)*m+98304*A-7350*B)*(a*(1+sin(f*x+e)))^m/(5+m)/(4+m)/(8+m)/(7+m)/(6+m 
)/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (159) = 318\).

Time = 0.15 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.09 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {{\left ({\left (B m^{4} + 22 \, B m^{3} + 179 \, B m^{2} + 638 \, B m + 840 \, B\right )} \cos \left (f x + e\right )^{8} - {\left ({\left (A + B\right )} m^{4} + {\left (17 \, A + 9 \, B\right )} m^{3} + 4 \, {\left (23 \, A + 5 \, B\right )} m^{2} + 160 \, A m\right )} \cos \left (f x + e\right )^{6} - 12 \, {\left ({\left (A + B\right )} m^{3} + {\left (11 \, A + 3 \, B\right )} m^{2} + 24 \, A m\right )} \cos \left (f x + e\right )^{4} - 96 \, {\left ({\left (A + B\right )} m^{2} + 8 \, A m\right )} \cos \left (f x + e\right )^{2} - 384 \, {\left (A + B\right )} m - {\left ({\left ({\left (A + B\right )} m^{4} + {\left (23 \, A + 15 \, B\right )} m^{3} + 2 \, {\left (97 \, A + 37 \, B\right )} m^{2} + 8 \, {\left (89 \, A + 15 \, B\right )} m + 960 \, A\right )} \cos \left (f x + e\right )^{6} + 12 \, {\left ({\left (A + B\right )} m^{3} + {\left (15 \, A + 7 \, B\right )} m^{2} + 4 \, {\left (17 \, A + 3 \, B\right )} m + 96 \, A\right )} \cos \left (f x + e\right )^{4} + 96 \, {\left ({\left (A + B\right )} m^{2} + 2 \, {\left (5 \, A + B\right )} m + 16 \, A\right )} \cos \left (f x + e\right )^{2} + 384 \, {\left (A + B\right )} m + 3072 \, A\right )} \sin \left (f x + e\right ) - 3072 \, A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{5} + 30 \, f m^{4} + 355 \, f m^{3} + 2070 \, f m^{2} + 5944 \, f m + 6720 \, f} \] Input:

integrate(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="f 
ricas")
 

Output:

-((B*m^4 + 22*B*m^3 + 179*B*m^2 + 638*B*m + 840*B)*cos(f*x + e)^8 - ((A + 
B)*m^4 + (17*A + 9*B)*m^3 + 4*(23*A + 5*B)*m^2 + 160*A*m)*cos(f*x + e)^6 - 
 12*((A + B)*m^3 + (11*A + 3*B)*m^2 + 24*A*m)*cos(f*x + e)^4 - 96*((A + B) 
*m^2 + 8*A*m)*cos(f*x + e)^2 - 384*(A + B)*m - (((A + B)*m^4 + (23*A + 15* 
B)*m^3 + 2*(97*A + 37*B)*m^2 + 8*(89*A + 15*B)*m + 960*A)*cos(f*x + e)^6 + 
 12*((A + B)*m^3 + (15*A + 7*B)*m^2 + 4*(17*A + 3*B)*m + 96*A)*cos(f*x + e 
)^4 + 96*((A + B)*m^2 + 2*(5*A + B)*m + 16*A)*cos(f*x + e)^2 + 384*(A + B) 
*m + 3072*A)*sin(f*x + e) - 3072*A)*(a*sin(f*x + e) + a)^m/(f*m^5 + 30*f*m 
^4 + 355*f*m^3 + 2070*f*m^2 + 5944*f*m + 6720*f)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**7*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1207 vs. \(2 (159) = 318\).

Time = 0.07 (sec) , antiderivative size = 1207, normalized size of antiderivative = 7.59 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \] Input:

integrate(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="m 
axima")
 

Output:

-(((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*a^m*sin(f* 
x + e)^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a^m*sin(f*x 
 + e)^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^m*sin(f*x + e)^5 + 3 
0*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + e)^4 - 120*(m^3 + 3*m^2 + 2*m 
)*a^m*sin(f*x + e)^3 + 360*(m^2 + m)*a^m*sin(f*x + e)^2 - 720*a^m*m*sin(f* 
x + e) + 720*a^m)*A*(sin(f*x + e) + 1)^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^ 
4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) - 3*((m^4 + 10*m^3 + 35*m^2 + 5 
0*m + 24)*a^m*sin(f*x + e)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + 
e)^4 - 4*(m^3 + 3*m^2 + 2*m)*a^m*sin(f*x + e)^3 + 12*(m^2 + m)*a^m*sin(f*x 
 + e)^2 - 24*a^m*m*sin(f*x + e) + 24*a^m)*A*(sin(f*x + e) + 1)^m/(m^5 + 15 
*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 3*((m^2 + 3*m + 2)*a^m*sin(f*x + 
e)^3 + (m^2 + m)*a^m*sin(f*x + e)^2 - 2*a^m*m*sin(f*x + e) + 2*a^m)*A*(sin 
(f*x + e) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) + ((m^7 + 28*m^6 + 322*m^5 + 196 
0*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*a^m*sin(f*x + e)^8 + (m^7 + 
 21*m^6 + 175*m^5 + 735*m^4 + 1624*m^3 + 1764*m^2 + 720*m)*a^m*sin(f*x + e 
)^7 - 7*(m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a^m*sin(f*x + 
e)^6 + 42*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^m*sin(f*x + e)^5 - 210 
*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + e)^4 + 840*(m^3 + 3*m^2 + 2*m) 
*a^m*sin(f*x + e)^3 - 2520*(m^2 + m)*a^m*sin(f*x + e)^2 + 5040*a^m*m*sin(f 
*x + e) - 5040*a^m)*B*(sin(f*x + e) + 1)^m/(m^8 + 36*m^7 + 546*m^6 + 45...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3184 vs. \(2 (159) = 318\).

Time = 0.18 (sec) , antiderivative size = 3184, normalized size of antiderivative = 20.03 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \] Input:

integrate(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="g 
iac")
 

Output:

-(((a*sin(f*x + e) + a)^m*m^6*sin(f*x + e)^7 + (a*sin(f*x + e) + a)^m*m^6* 
sin(f*x + e)^6 + 21*(a*sin(f*x + e) + a)^m*m^5*sin(f*x + e)^7 + 15*(a*sin( 
f*x + e) + a)^m*m^5*sin(f*x + e)^6 + 175*(a*sin(f*x + e) + a)^m*m^4*sin(f* 
x + e)^7 - 6*(a*sin(f*x + e) + a)^m*m^5*sin(f*x + e)^5 + 85*(a*sin(f*x + e 
) + a)^m*m^4*sin(f*x + e)^6 + 735*(a*sin(f*x + e) + a)^m*m^3*sin(f*x + e)^ 
7 - 60*(a*sin(f*x + e) + a)^m*m^4*sin(f*x + e)^5 + 225*(a*sin(f*x + e) + a 
)^m*m^3*sin(f*x + e)^6 + 1624*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^7 + 
30*(a*sin(f*x + e) + a)^m*m^4*sin(f*x + e)^4 - 210*(a*sin(f*x + e) + a)^m* 
m^3*sin(f*x + e)^5 + 274*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^6 + 1764* 
(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^7 + 180*(a*sin(f*x + e) + a)^m*m^3*s 
in(f*x + e)^4 - 300*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^5 + 120*(a*sin 
(f*x + e) + a)^m*m*sin(f*x + e)^6 + 720*(a*sin(f*x + e) + a)^m*sin(f*x + e 
)^7 - 120*(a*sin(f*x + e) + a)^m*m^3*sin(f*x + e)^3 + 330*(a*sin(f*x + e) 
+ a)^m*m^2*sin(f*x + e)^4 - 144*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^5 - 
360*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^3 + 180*(a*sin(f*x + e) + a)^m 
*m*sin(f*x + e)^4 + 360*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^2 - 240*(a 
*sin(f*x + e) + a)^m*m*sin(f*x + e)^3 + 360*(a*sin(f*x + e) + a)^m*m*sin(f 
*x + e)^2 - 720*(a*sin(f*x + e) + a)^m*m*sin(f*x + e) + 720*(a*sin(f*x + e 
) + a)^m)*A/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13 
068*m + 5040) - 3*((a*sin(f*x + e) + a)^m*m^4*sin(f*x + e)^5 + (a*sin(f...
 

Mupad [B] (verification not implemented)

Time = 44.05 (sec) , antiderivative size = 783, normalized size of antiderivative = 4.92 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx =\text {Too large to display} \] Input:

int(cos(e + f*x)^7*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m,x)
 

Output:

-exp(- e*8i - f*x*8i)*(a + a*sin(e + f*x))^m*((exp(e*8i + f*x*8i)*cos(4*e 
+ 4*f*x)*(23520*B - 8448*A*m + 17864*B*m - 4320*A*m^2 - 600*A*m^3 - 24*A*m 
^4 + 3956*B*m^2 + 208*B*m^3 + 4*B*m^4))/(128*f*(5944*m + 2070*m^2 + 355*m^ 
3 + 30*m^4 + m^5 + 6720)) - (exp(e*8i + f*x*8i)*(786432*A - 58800*B + 2370 
56*A*m + 53644*B*m + 32320*A*m^2 + 2512*A*m^3 + 80*A*m^4 + 4814*B*m^2 + 33 
2*B*m^3 + 10*B*m^4))/(256*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 
6720)) - (exp(e*8i + f*x*8i)*cos(2*e + 2*f*x)*(77184*A*m - 47040*B - 35728 
*B*m + 20112*A*m^2 + 1788*A*m^3 + 60*A*m^4 - 376*B*m^2 + 76*B*m^3 + 4*B*m^ 
4))/(128*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720)) + (B*exp(e 
*8i + f*x*8i)*cos(8*e + 8*f*x)*(638*m + 179*m^2 + 22*m^3 + m^4 + 840))/(12 
8*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720)) + (exp(e*8i + f*x 
*8i)*sin(5*e + 5*f*x)*(A*8i + A*m*1i + B*m*1i)*(706*m + 123*m^2 + 5*m^3 + 
1176)*1i)/(64*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720)) + (ex 
p(e*8i + f*x*8i)*sin(3*e + 3*f*x)*(A*8i + A*m*1i + B*m*1i)*(1070*m + 93*m^ 
2 + 3*m^3 + 1960)*3i)/(64*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 
6720)) + (exp(e*8i + f*x*8i)*cos(6*e + 6*f*x)*(9*m + m^2 + 20)*(84*B - 8*A 
*m + 26*B*m - A*m^2 + B*m^2))/(32*f*(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 
+ m^5 + 6720)) + (exp(e*8i + f*x*8i)*sin(7*e + 7*f*x)*(A*8i + A*m*1i + B*m 
*1i)*(74*m + 15*m^2 + m^3 + 120)*1i)/(64*f*(5944*m + 2070*m^2 + 355*m^3 + 
30*m^4 + m^5 + 6720)) + (exp(e*8i + f*x*8i)*sin(e + f*x)*(A*8i + A*m*1i...
 

Reduce [F]

\[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \cos \left (f x +e \right )^{7} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )d x \] Input:

int(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)
 

Output:

int(cos(f*x+e)^7*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)