\(\int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1020]

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2915, 78} \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {B (a \sin (e+f x)+a)^{m+8}}{a^8 f (m+8)}-\frac {(A-7 B) (a \sin (e+f x)+a)^{m+7}}{a^7 f (m+7)}+\frac {6 (A-3 B) (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}-\frac {4 (3 A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac {8 (A-B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)} \]

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^3 (a+x)^{3+m} \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^7 f} \\ & = \frac {\text {Subst}\left (\int \left (8 a^3 (A-B) (a+x)^{3+m}-4 a^2 (3 A-5 B) (a+x)^{4+m}+6 a (A-3 B) (a+x)^{5+m}+(-A+7 B) (a+x)^{6+m}-\frac {B (a+x)^{7+m}}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^7 f} \\ & = \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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.83 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {(a (1+\sin (e+f x)))^{4+m} \left (\frac {8 a^4 (A-B)}{4+m}-\frac {4 a^4 (3 A-5 B) (1+\sin (e+f x))}{5+m}+\frac {6 a^4 (A-3 B) (1+\sin (e+f x))^2}{6+m}-\frac {a^4 (A-7 B) (1+\sin (e+f x))^3}{7+m}-\frac {B (a+a \sin (e+f x))^4}{8+m}\right )}{a^8 f} \]

[In]

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

[Out]

((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)

Maple [B] (verified)

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

Time = 14.80 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.43

method	result	size
parallelrisch	\(\frac {\left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (\left (\left (15 A +B \right ) m^{4}+\left (447 A +19 B \right ) m^{3}+\left (5028 A -94 B \right ) m^{2}+\left (19296 A -8932 B \right ) m -11760 B \right ) \cos \left (2 f x +2 e \right )+\left (\left (6 A -B \right ) m^{4}+\left (150 A -52 B \right ) m^{3}+\left (1080 A -989 B \right ) m^{2}+\left (2112 A -4466 B \right ) m -5880 B \right ) \cos \left (4 f x +4 e \right )+\left (5+m \right ) \left (4+m \right ) \left (\left (A -B \right ) m^{2}+\left (8 A -26 B \right ) m -84 B \right ) \cos \left (6 f x +6 e \right )+\frac {9 \left (\left (A +B \right ) m +8 A \right ) \left (m^{3}+31 m^{2}+\frac {1070}{3} m +\frac {1960}{3}\right ) \sin \left (3 f x +3 e \right )}{2}+\frac {5 \left (\left (A +B \right ) m +8 A \right ) \left (4+m \right ) \left (m^{2}+\frac {103}{5} m +\frac {294}{5}\right ) \sin \left (5 f x +5 e \right )}{2}+\frac {\left (\left (A +B \right ) m +8 A \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \sin \left (7 f x +7 e \right )}{2}-\frac {B \left (5+m \right ) \left (4+m \right ) \left (7+m \right ) \left (6+m \right ) \cos \left (8 f x +8 e \right )}{4}+\frac {5 \left (\left (A +B \right ) m +8 A \right ) \left (m^{3}+\frac {171}{5} m^{2}+\frac {2578}{5} m +5880\right ) \sin \left (f x +e \right )}{2}+\left (10 A +\frac {5 B}{4}\right ) m^{4}+\left (\frac {83 B}{2}+314 A \right ) m^{3}+\left (\frac {2407 B}{4}+4040 A \right ) m^{2}+\left (29632 A +\frac {13411 B}{2}\right ) m +98304 A -7350 B \right )}{32 \left (5+m \right ) \left (4+m \right ) \left (8+m \right ) \left (7+m \right ) \left (6+m \right ) f}\)	\(387\)
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 \left (\sin ^{8}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{4}\left (f x +e \right )\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 (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 ) \left (\sin ^{2}\left (f x +e \right )\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 {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 ) \left (\sin ^{3}\left (f x +e \right )\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 +B m +8 A \right ) \left (\sin ^{7}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{6}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{5}\left (f x +e \right )\right ) {\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 \left (\sin ^{8}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{4}\left (f x +e \right )\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 (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 ) \left (\sin ^{2}\left (f x +e \right )\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 {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 ) \left (\sin ^{3}\left (f x +e \right )\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 +B m +8 A \right ) \left (\sin ^{7}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{6}\left (f x +e \right )\right ) {\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 ) \left (\sin ^{5}\left (f x +e \right )\right ) {\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 )}\)	\(707\)

[In]

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

[Out]

1/32*(a*(1+sin(f*x+e)))^m*(((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)+(5+m)*(4+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*((A+B)*m+8*A)*(4+m)*(m^2+103/5*m+294/5)*sin(5*f*x+5*e)+1/2*((A+B)*m+8*A)*(6+m)*(5+m)*(4+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)+(10*A+5/
4*B)*m^4+(83/2*B+314*A)*m^3+(2407/4*B+4040*A)*m^2+(29632*A+13411/2*B)*m+98304*A-7350*B)/(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 = 0.35 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.09 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {{\left ({\left (B m^{4} + 22 \, B m^{3} + 179 \, B m^{2} + 638 \, B m + 840 \, B\right )} \cos \left (f x + e\right )^{8} - {\left ({\left (A + B\right )} m^{4} + {\left (17 \, A + 9 \, B\right )} m^{3} + 4 \, {\left (23 \, A + 5 \, B\right )} m^{2} + 160 \, A m\right )} \cos \left (f x + e\right )^{6} - 12 \, {\left ({\left (A + B\right )} m^{3} + {\left (11 \, A + 3 \, B\right )} m^{2} + 24 \, A m\right )} \cos \left (f x + e\right )^{4} - 96 \, {\left ({\left (A + B\right )} m^{2} + 8 \, A m\right )} \cos \left (f x + e\right )^{2} - 384 \, {\left (A + B\right )} m - {\left ({\left ({\left (A + B\right )} m^{4} + {\left (23 \, A + 15 \, B\right )} m^{3} + 2 \, {\left (97 \, A + 37 \, B\right )} m^{2} + 8 \, {\left (89 \, A + 15 \, B\right )} m + 960 \, A\right )} \cos \left (f x + e\right )^{6} + 12 \, {\left ({\left (A + B\right )} m^{3} + {\left (15 \, A + 7 \, B\right )} m^{2} + 4 \, {\left (17 \, A + 3 \, B\right )} m + 96 \, A\right )} \cos \left (f x + e\right )^{4} + 96 \, {\left ({\left (A + B\right )} m^{2} + 2 \, {\left (5 \, A + B\right )} m + 16 \, A\right )} \cos \left (f x + e\right )^{2} + 384 \, {\left (A + B\right )} m + 3072 \, A\right )} \sin \left (f x + e\right ) - 3072 \, A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{5} + 30 \, f m^{4} + 355 \, f m^{3} + 2070 \, f m^{2} + 5944 \, f m + 6720 \, f} \]

[In]

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

[Out]

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

[In]

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

[Out]

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 = 0.26 (sec) , antiderivative size = 1207, normalized size of antiderivative = 7.59 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(((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 +
30*(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 + 196
0*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040) - 3*((m^4 + 10*m^3 + 35*m^2 + 50*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*s
in(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 + 1960*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*s
in(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*(s
in(f*x + e) + 1)^m/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)
- 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*a^m*sin(f*x + e)^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 2
4*m)*a^m*sin(f*x + e)^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^m*sin(f*x + e)^4 + 20*(m^3 + 3*m^2 + 2*m)*a^m*sin(f
*x + e)^3 - 60*(m^2 + m)*a^m*sin(f*x + e)^2 + 120*a^m*m*sin(f*x + e) - 120*a^m)*B*(sin(f*x + e) + 1)^m/(m^6 +
21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720) + 3*((m^3 + 6*m^2 + 11*m + 6)*a^m*sin(f*x + e)^4 + (m^3
+ 3*m^2 + 2*m)*a^m*sin(f*x + e)^3 - 3*(m^2 + m)*a^m*sin(f*x + e)^2 + 6*a^m*m*sin(f*x + e) - 6*a^m)*B*(sin(f*x
+ e) + 1)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) - (a^m*(m + 1)*sin(f*x + e)^2 + a^m*m*sin(f*x + e) - a^m)*B*(s
in(f*x + e) + 1)^m/(m^2 + 3*m + 2) - (a*sin(f*x + e) + a)^(m + 1)*A/(a*(m + 1)))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1320 vs. \(2 (159) = 318\).

Time = 0.38 (sec) , antiderivative size = 1320, normalized size of antiderivative = 8.30 \[ \int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(((a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m*m^3 - 6*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*a*m^3 +
12*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^2*m^3 - 8*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^
3*m^3 + 15*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m*m^2 - 96*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^
m*a*m^2 + 204*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^2*m^2 - 144*(a*sin(f*x + e) + a)^4*(a*sin(f*x +
e) + a)^m*a^3*m^2 + 74*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m*m - 498*(a*sin(f*x + e) + a)^6*(a*sin(f*x
+ e) + a)^m*a*m + 1128*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^2*m - 856*(a*sin(f*x + e) + a)^4*(a*si
n(f*x + e) + a)^m*a^3*m + 120*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m - 840*(a*sin(f*x + e) + a)^6*(a*si
n(f*x + e) + a)^m*a + 2016*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^2 - 1680*(a*sin(f*x + e) + a)^4*(a*
sin(f*x + e) + a)^m*a^3)*A/(a^6*m^4 + 22*a^6*m^3 + 179*a^6*m^2 + 638*a^6*m + 840*a^6) + ((a*sin(f*x + e) + a)^
8*(a*sin(f*x + e) + a)^m*m^4 - 7*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m*a*m^4 + 18*(a*sin(f*x + e) + a)
^6*(a*sin(f*x + e) + a)^m*a^2*m^4 - 20*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^3*m^4 + 8*(a*sin(f*x +
e) + a)^4*(a*sin(f*x + e) + a)^m*a^4*m^4 + 22*(a*sin(f*x + e) + a)^8*(a*sin(f*x + e) + a)^m*m^3 - 161*(a*sin(f
*x + e) + a)^7*(a*sin(f*x + e) + a)^m*a*m^3 + 432*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*a^2*m^3 - 500*
(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^3*m^3 + 208*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^4*
m^3 + 179*(a*sin(f*x + e) + a)^8*(a*sin(f*x + e) + a)^m*m^2 - 1358*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)
^m*a*m^2 + 3798*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*a^2*m^2 - 4600*(a*sin(f*x + e) + a)^5*(a*sin(f*x
+ e) + a)^m*a^3*m^2 + 2008*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^4*m^2 + 638*(a*sin(f*x + e) + a)^8
*(a*sin(f*x + e) + a)^m*m - 4984*(a*sin(f*x + e) + a)^7*(a*sin(f*x + e) + a)^m*a*m + 14472*(a*sin(f*x + e) + a
)^6*(a*sin(f*x + e) + a)^m*a^2*m - 18400*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^3*m + 8528*(a*sin(f*x
+ e) + a)^4*(a*sin(f*x + e) + a)^m*a^4*m + 840*(a*sin(f*x + e) + a)^8*(a*sin(f*x + e) + a)^m - 6720*(a*sin(f*
x + e) + a)^7*(a*sin(f*x + e) + a)^m*a + 20160*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*a^2 - 26880*(a*si
n(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a^3 + 13440*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^4)*B/((a^
6*m^5 + 30*a^6*m^4 + 355*a^6*m^3 + 2070*a^6*m^2 + 5944*a^6*m + 6720*a^6)*a))/(a*f)

Mupad [B] (verification not implemented)

Time = 18.18 (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=-{\mathrm {e}}^{-e\,8{}\mathrm {i}-f\,x\,8{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (-\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (786432\,A-58800\,B+237056\,A\,m+53644\,B\,m+32320\,A\,m^2+2512\,A\,m^3+80\,A\,m^4+4814\,B\,m^2+332\,B\,m^3+10\,B\,m^4\right )}{256\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\left (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\right )}{128\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}-\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (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\right )}{128\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {B\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (8\,e+8\,f\,x\right )\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}{128\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\left (A\,8{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (5\,m^3+123\,m^2+706\,m+1176\right )\,1{}\mathrm {i}}{64\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\left (A\,8{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (3\,m^3+93\,m^2+1070\,m+1960\right )\,3{}\mathrm {i}}{64\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\left (m^2+9\,m+20\right )\,\left (84\,B-8\,A\,m+26\,B\,m-A\,m^2+B\,m^2\right )}{32\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\left (A\,8{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^3+15\,m^2+74\,m+120\right )\,1{}\mathrm {i}}{64\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (A\,8{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (5\,m^3+171\,m^2+2578\,m+29400\right )\,1{}\mathrm {i}}{64\,f\,\left (m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720\right )}\right ) \]

[In]

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

[Out]

-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 + 237056*A*m + 53644*B*m + 32320*A*m^2 +
2512*A*m^3 + 80*A*m^4 + 4814*B*m^2 + 332*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))/(128*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)) + (exp(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 + B*m*1i)*(2578*m + 171*m^2 + 5*m^3 + 29400)*1i)/(64*f*(5944*m
+ 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720)))