Integrand size = 31, antiderivative size = 123 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {4 (A-B) (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}-\frac {4 (A-2 B) (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac {(A-5 B) (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}+\frac {B (a+a \sin (e+f x))^{6+m}}{a^6 f (6+m)} \]
4*(A-B)*(a+a*sin(f*x+e))^(3+m)/a^3/f/(3+m)-4*(A-2*B)*(a+a*sin(f*x+e))^(4+m )/a^4/f/(4+m)+(A-5*B)*(a+a*sin(f*x+e))^(5+m)/a^5/f/(5+m)+B*(a+a*sin(f*x+e) )^(6+m)/a^6/f/(6+m)
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {(a (1+\sin (e+f x)))^{3+m} \left (\frac {4 a^3 (A-B)}{3+m}-\frac {4 a^3 (A-2 B) (1+\sin (e+f x))}{4+m}+\frac {a^3 (A-5 B) (1+\sin (e+f x))^2}{5+m}+\frac {B (a+a \sin (e+f x))^3}{6+m}\right )}{a^6 f} \]
((a*(1 + Sin[e + f*x]))^(3 + m)*((4*a^3*(A - B))/(3 + m) - (4*a^3*(A - 2*B )*(1 + Sin[e + f*x]))/(4 + m) + (a^3*(A - 5*B)*(1 + Sin[e + f*x])^2)/(5 + m) + (B*(a + a*Sin[e + f*x])^3)/(6 + m)))/(a^6*f)
Time = 0.33 (sec) , antiderivative size = 113, 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.
| \(\displaystyle \int \cos ^5(e+f x) (a \sin (e+f x)+a)^m (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x)^5 (a \sin (e+f x)+a)^m (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (e+f x))^2 (\sin (e+f x) a+a)^{m+2} (a A+a B \sin (e+f x))}{a}d(a \sin (e+f x))}{a^5 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a-a \sin (e+f x))^2 (\sin (e+f x) a+a)^{m+2} (a A+a B \sin (e+f x))d(a \sin (e+f x))}{a^6 f}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (4 a^3 (A-B) (\sin (e+f x) a+a)^{m+2}-4 a^2 (A-2 B) (\sin (e+f x) a+a)^{m+3}+a (A-5 B) (\sin (e+f x) a+a)^{m+4}+B (\sin (e+f x) a+a)^{m+5}\right )d(a \sin (e+f x))}{a^6 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {4 a^3 (A-B) (a \sin (e+f x)+a)^{m+3}}{m+3}-\frac {4 a^2 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{m+4}+\frac {a (A-5 B) (a \sin (e+f x)+a)^{m+5}}{m+5}+\frac {B (a \sin (e+f x)+a)^{m+6}}{m+6}}{a^6 f}\) |
((4*a^3*(A - B)*(a + a*Sin[e + f*x])^(3 + m))/(3 + m) - (4*a^2*(A - 2*B)*( a + a*Sin[e + f*x])^(4 + m))/(4 + m) + (a*(A - 5*B)*(a + a*Sin[e + f*x])^( 5 + m))/(5 + m) + (B*(a + a*Sin[e + f*x])^(6 + m))/(6 + m))/(a^6*f)
3.11.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(123)=246\).
Time = 6.23 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.07
| method | result | size |
| parallelrisch | \(\frac {\left (\left (\left (4 A +\frac {B}{4}\right ) m^{3}+\left (68 A -B \right ) m^{2}+\left (264 A -\frac {705 B}{4}\right ) m -225 B \right ) \cos \left (2 f x +2 e \right )+\left (\left (A -\frac {B}{2}\right ) m^{2}+\left (6 A -\frac {27 B}{2}\right ) m -30 B \right ) \left (3+m \right ) \cos \left (4 f x +4 e \right )+\frac {3 \left (m^{2}+\frac {53}{3} m +\frac {100}{3}\right ) \left (\left (A +B \right ) m +6 A \right ) \sin \left (3 f x +3 e \right )}{2}+\frac {\left (4+m \right ) \left (\left (A +B \right ) m +6 A \right ) \left (3+m \right ) \sin \left (5 f x +5 e \right )}{2}-\frac {B \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \cos \left (6 f x +6 e \right )}{4}+\left (m^{2}+23 m +300\right ) \left (\left (A +B \right ) m +6 A \right ) \sin \left (f x +e \right )+\left (\frac {B}{2}+3 A \right ) m^{3}+\left (11 B +59 A \right ) m^{2}+\left (502 A +\frac {277 B}{2}\right ) m +1536 A -150 B \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m}}{8 \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (6+m \right ) f}\) | \(255\) |
| derivativedivides | \(\frac {\left (A \,m^{3}+17 A \,m^{2}-B \,m^{2}+98 A m -15 B m +192 A -60 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}+\frac {B \left (\sin ^{6}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (6+m \right )}+\frac {\left (A m +B m +6 A \right ) \left (\sin ^{5}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+11 m +30\right )}+\frac {\left (A \,m^{2}-2 B \,m^{2}+6 A m -27 B m -60 B \right ) \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+15 m^{2}+74 m +120\right )}+\frac {\left (A \,m^{3}+B \,m^{3}+21 A \,m^{2}+15 B \,m^{2}+150 A m +60 B m +360 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {\left (2 A \,m^{3}-B \,m^{3}+26 A \,m^{2}-22 B \,m^{2}+84 A m -141 B m -180 B \right ) \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {2 \left (A \,m^{3}+B \,m^{3}+17 A \,m^{2}+11 B \,m^{2}+86 A m +20 B m +120 A \right ) \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}\) | \(456\) |
| default | \(\frac {\left (A \,m^{3}+17 A \,m^{2}-B \,m^{2}+98 A m -15 B m +192 A -60 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}+\frac {B \left (\sin ^{6}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (6+m \right )}+\frac {\left (A m +B m +6 A \right ) \left (\sin ^{5}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+11 m +30\right )}+\frac {\left (A \,m^{2}-2 B \,m^{2}+6 A m -27 B m -60 B \right ) \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+15 m^{2}+74 m +120\right )}+\frac {\left (A \,m^{3}+B \,m^{3}+21 A \,m^{2}+15 B \,m^{2}+150 A m +60 B m +360 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {\left (2 A \,m^{3}-B \,m^{3}+26 A \,m^{2}-22 B \,m^{2}+84 A m -141 B m -180 B \right ) \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {2 \left (A \,m^{3}+B \,m^{3}+17 A \,m^{2}+11 B \,m^{2}+86 A m +20 B m +120 A \right ) \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}\) | \(456\) |
1/8*(((4*A+1/4*B)*m^3+(68*A-B)*m^2+(264*A-705/4*B)*m-225*B)*cos(2*f*x+2*e) +((A-1/2*B)*m^2+(6*A-27/2*B)*m-30*B)*(3+m)*cos(4*f*x+4*e)+3/2*(m^2+53/3*m+ 100/3)*((A+B)*m+6*A)*sin(3*f*x+3*e)+1/2*(4+m)*((A+B)*m+6*A)*(3+m)*sin(5*f* x+5*e)-1/4*B*(5+m)*(4+m)*(3+m)*cos(6*f*x+6*e)+(m^2+23*m+300)*((A+B)*m+6*A) *sin(f*x+e)+(1/2*B+3*A)*m^3+(11*B+59*A)*m^2+(502*A+277/2*B)*m+1536*A-150*B )*(a*(1+sin(f*x+e)))^m/(5+m)/(4+m)/(3+m)/(6+m)/f
Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.80 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {{\left ({\left (B m^{3} + 12 \, B m^{2} + 47 \, B m + 60 \, B\right )} \cos \left (f x + e\right )^{6} - {\left ({\left (A + B\right )} m^{3} + 3 \, {\left (3 \, A + B\right )} m^{2} + 18 \, A m\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left ({\left (A + B\right )} m^{2} + 6 \, A m\right )} \cos \left (f x + e\right )^{2} - 32 \, {\left (A + B\right )} m - {\left ({\left ({\left (A + B\right )} m^{3} + {\left (13 \, A + 7 \, B\right )} m^{2} + 6 \, {\left (9 \, A + 2 \, B\right )} m + 72 \, A\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left ({\left (A + B\right )} m^{2} + 2 \, {\left (4 \, A + B\right )} m + 12 \, A\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (A + B\right )} m + 192 \, A\right )} \sin \left (f x + e\right ) - 192 \, A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{4} + 18 \, f m^{3} + 119 \, f m^{2} + 342 \, f m + 360 \, f} \]
-((B*m^3 + 12*B*m^2 + 47*B*m + 60*B)*cos(f*x + e)^6 - ((A + B)*m^3 + 3*(3* A + B)*m^2 + 18*A*m)*cos(f*x + e)^4 - 8*((A + B)*m^2 + 6*A*m)*cos(f*x + e) ^2 - 32*(A + B)*m - (((A + B)*m^3 + (13*A + 7*B)*m^2 + 6*(9*A + 2*B)*m + 7 2*A)*cos(f*x + e)^4 + 8*((A + B)*m^2 + 2*(4*A + B)*m + 12*A)*cos(f*x + e)^ 2 + 32*(A + B)*m + 192*A)*sin(f*x + e) - 192*A)*(a*sin(f*x + e) + a)^m/(f* m^4 + 18*f*m^3 + 119*f*m^2 + 342*f*m + 360*f)
Leaf count of result is larger than twice the leaf count of optimal. 22522 vs. \(2 (107) = 214\).
Time = 95.27 (sec) , antiderivative size = 22522, normalized size of antiderivative = 183.11 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]
Piecewise((x*(A + B*sin(e))*(a*sin(e) + a)**m*cos(e)**5, Eq(f, 0)), (-32*A *sin(e + f*x)**4/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*sin( e + f*x) + 60*a**6*f) - 100*A*sin(e + f*x)**3/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin( e + f*x)**2 + 300*a**6*f*sin(e + f*x) + 60*a**6*f) + 16*A*sin(e + f*x)**2* cos(e + f*x)**2/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*sin(e + f*x) + 60*a**6*f) - 116*A*sin(e + f*x)**2/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*sin(e + f*x) + 60*a**6*f) + 20*A*sin(e + f*x)*cos( e + f*x)**2/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600* a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*sin(e + f *x) + 60*a**6*f) - 60*A*sin(e + f*x)/(60*a**6*f*sin(e + f*x)**5 + 300*a**6 *f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)* *2 + 300*a**6*f*sin(e + f*x) + 60*a**6*f) - 12*A*cos(e + f*x)**4/(60*a**6* f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)** 3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*sin(e + f*x) + 60*a**6*f) + 4* A*cos(e + f*x)**2/(60*a**6*f*sin(e + f*x)**5 + 300*a**6*f*sin(e + f*x)**4 + 600*a**6*f*sin(e + f*x)**3 + 600*a**6*f*sin(e + f*x)**2 + 300*a**6*f*...
Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (123) = 246\).
Time = 0.23 (sec) , antiderivative size = 643, normalized size of antiderivative = 5.23 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {\frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (f x + e\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 24 \, a^{m} m \sin \left (f x + e\right ) + 24 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} a^{m} \sin \left (f x + e\right )^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (f x + e\right )^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 60 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 120 \, a^{m} m \sin \left (f x + e\right ) - 120 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} - \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} A}{a {\left (m + 1\right )}}}{f} \]
(((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*sin(f*x + e)^2 - 24*a^m*m*sin(f*x + e) + 24*a^m)*A*(s in(f*x + e) + 1)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) - 2*((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^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 + 24*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) - 2*((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*(sin(f*x + e) + 1)^m/(m^2 + 3*m + 2) + (a*sin(f*x + e) + a)^(m + 1)*A/(a*(m + 1)))/f
Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (123) = 246\).
Time = 0.34 (sec) , antiderivative size = 811, normalized size of antiderivative = 6.59 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]
(((a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m^2 - 4*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m^2 + 4*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m^2 + 7*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*m - 32 *(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a*m + 36*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2*m + 12*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m - 60*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a + 80*(a*si n(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^2)*A/(a^4*m^3 + 12*a^4*m^2 + 47 *a^4*m + 60*a^4) + ((a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*m^3 - 5* (a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m^3 + 8*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m^3 - 4*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a^3*m^3 + 12*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*m^2 - 65*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m^2 + 112*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m^2 - 60*(a*sin(f*x + e) + a)^3*(a *sin(f*x + e) + a)^m*a^3*m^2 + 47*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e) + a)^m*m - 270*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a*m + 504*(a*s in(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2*m - 296*(a*sin(f*x + e) + a) ^3*(a*sin(f*x + e) + a)^m*a^3*m + 60*(a*sin(f*x + e) + a)^6*(a*sin(f*x + e ) + a)^m - 360*(a*sin(f*x + e) + a)^5*(a*sin(f*x + e) + a)^m*a + 720*(a*si n(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*a^2 - 480*(a*sin(f*x + e) + a)^3* (a*sin(f*x + e) + a)^m*a^3)*B/((a^4*m^4 + 18*a^4*m^3 + 119*a^4*m^2 + 34...
Time = 17.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.20 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (12288\,A-1200\,B+4016\,A\,m+1108\,B\,m+472\,A\,m^2+24\,A\,m^3+88\,B\,m^2+4\,B\,m^3\right )}{64\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (1056\,A\,m-900\,B-705\,B\,m+272\,A\,m^2+16\,A\,m^3-4\,B\,m^2+B\,m^3\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\left (m+3\right )\,\left (60\,B-12\,A\,m+27\,B\,m-2\,A\,m^2+B\,m^2\right )}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+23\,m+300\right )\,1{}\mathrm {i}}{8\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+7\,m+12\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {B\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\left (m^3+12\,m^2+47\,m+60\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (3\,m^2+53\,m+100\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}\right ) \]
-exp(- e*6i - f*x*6i)*(a + a*sin(e + f*x))^m*((exp(e*6i + f*x*6i)*cos(4*e + 4*f*x)*(m + 3)*(60*B - 12*A*m + 27*B*m - 2*A*m^2 + B*m^2))/(16*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) - (exp(e*6i + f*x*6i)*cos(2*e + 2*f*x)*(1 056*A*m - 900*B - 705*B*m + 272*A*m^2 + 16*A*m^3 - 4*B*m^2 + B*m^3))/(32*f *(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) - (exp(e*6i + f*x*6i)*(12288*A - 1200*B + 4016*A*m + 1108*B*m + 472*A*m^2 + 24*A*m^3 + 88*B*m^2 + 4*B*m^3)) /(64*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f*x*6i)*sin(e + f*x)*(A*6i + A*m*1i + B*m*1i)*(23*m + m^2 + 300)*1i)/(8*f*(342*m + 119* m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f*x*6i)*sin(5*e + 5*f*x)*(A*6i + A*m*1i + B*m*1i)*(7*m + m^2 + 12)*1i)/(16*f*(342*m + 119*m^2 + 18*m^3 + m^ 4 + 360)) + (B*exp(e*6i + f*x*6i)*cos(6*e + 6*f*x)*(47*m + 12*m^2 + m^3 + 60))/(32*f*(342*m + 119*m^2 + 18*m^3 + m^4 + 360)) + (exp(e*6i + f*x*6i)*s in(3*e + 3*f*x)*(A*6i + A*m*1i + B*m*1i)*(53*m + 3*m^2 + 100)*1i)/(16*f*(3 42*m + 119*m^2 + 18*m^3 + m^4 + 360)))