Integrand size = 31, antiderivative size = 133 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {(c-d)^3 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac {3 (c-d)^2 d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac {3 (c-d) d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}+\frac {d^3 (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)} \]
(c-d)^3*(a+a*sin(f*x+e))^(1+m)/a/f/(1+m)+3*(c-d)^2*d*(a+a*sin(f*x+e))^(2+m )/a^2/f/(2+m)+3*(c-d)*d^2*(a+a*sin(f*x+e))^(3+m)/a^3/f/(3+m)+d^3*(a+a*sin( f*x+e))^(4+m)/a^4/f/(4+m)
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {(a (1+\sin (e+f x)))^{1+m} \left (\frac {a^3 (c-d)^3}{1+m}+\frac {3 a^3 (c-d)^2 d (1+\sin (e+f x))}{2+m}+\frac {3 a^3 (c-d) d^2 (1+\sin (e+f x))^2}{3+m}+\frac {d^3 (a+a \sin (e+f x))^3}{4+m}\right )}{a^4 f} \]
((a*(1 + Sin[e + f*x]))^(1 + m)*((a^3*(c - d)^3)/(1 + m) + (3*a^3*(c - d)^ 2*d*(1 + Sin[e + f*x]))/(2 + m) + (3*a^3*(c - d)*d^2*(1 + Sin[e + f*x])^2) /(3 + m) + (d^3*(a + a*Sin[e + f*x])^3)/(4 + m)))/(a^4*f)
Time = 0.34 (sec) , antiderivative size = 123, 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, 3312, 27, 53, 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.
| \(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m (a c+a d \sin (e+f x))^3}{a^3}d(a \sin (e+f x))}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (a c+a d \sin (e+f x))^3d(a \sin (e+f x))}{a^4 f}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\int \left (a^3 (c-d)^3 (\sin (e+f x) a+a)^m+3 a^2 (c-d)^2 d (\sin (e+f x) a+a)^{m+1}+3 a (c-d) d^2 (\sin (e+f x) a+a)^{m+2}+d^3 (\sin (e+f x) a+a)^{m+3}\right )d(a \sin (e+f x))}{a^4 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^3 (c-d)^3 (a \sin (e+f x)+a)^{m+1}}{m+1}+\frac {3 a^2 d (c-d)^2 (a \sin (e+f x)+a)^{m+2}}{m+2}+\frac {3 a d^2 (c-d) (a \sin (e+f x)+a)^{m+3}}{m+3}+\frac {d^3 (a \sin (e+f x)+a)^{m+4}}{m+4}}{a^4 f}\) |
((a^3*(c - d)^3*(a + a*Sin[e + f*x])^(1 + m))/(1 + m) + (3*a^2*(c - d)^2*d *(a + a*Sin[e + f*x])^(2 + m))/(2 + m) + (3*a*(c - d)*d^2*(a + a*Sin[e + f *x])^(3 + m))/(3 + m) + (d^3*(a + a*Sin[e + f*x])^(4 + m))/(4 + m))/(a^4*f )
3.10.21.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 3.40 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.97
| method | result | size |
| parallelrisch | \(\frac {\left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (-\frac {3 \left (1+m \right ) \left (\left (\frac {1}{3} m^{2}+\frac {2}{3} m +2\right ) d^{2}+c m \left (4+m \right ) d +c^{2} \left (4+m \right ) \left (3+m \right )\right ) d \cos \left (2 f x +2 e \right )}{2}+\frac {d^{3} \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \cos \left (4 f x +4 e \right )}{8}-\frac {3 \left (1+m \right ) \left (\frac {d m}{3}+c \left (4+m \right )\right ) d^{2} \left (2+m \right ) \sin \left (3 f x +3 e \right )}{4}+\left (\frac {3 m \left (m^{2}+3 m +10\right ) d^{3}}{4}+\frac {9 c \left (4+m \right ) \left (m^{2}+\frac {1}{3} m +2\right ) d^{2}}{4}+3 c^{2} m \left (4+m \right ) \left (3+m \right ) d +c^{3} \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right ) \sin \left (f x +e \right )+\frac {3 \left (-1+m \right ) \left (m^{2}+3 m +10\right ) d^{3}}{8}+\frac {3 c \left (4+m \right ) \left (m^{2}+m +4\right ) d^{2}}{2}+\frac {3 c^{2} \left (-1+m \right ) \left (4+m \right ) \left (3+m \right ) d}{2}+c^{3} \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right )}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(262\) |
| derivativedivides | \(\frac {\left (c^{3} m^{3}+9 c^{3} m^{2}-3 c^{2} d \,m^{2}+26 c^{3} m -21 c^{2} d m +6 c \,d^{2} m +24 c^{3}-36 c^{2} d +24 d^{2} c -6 d^{3}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {d^{3} \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (4+m \right )}+\frac {\left (c^{3} m^{3}+3 c^{2} d \,m^{3}+9 c^{3} m^{2}+21 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+26 c^{3} m +36 c^{2} d m -24 c \,d^{2} m +6 d^{3} m +24 c^{3}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 c m +d m +12 c \right ) d^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+7 m +12\right )}+\frac {3 \left (c^{2} m^{2}+c d \,m^{2}+7 c^{2} m +4 c d m -d^{2} m +12 c^{2}\right ) d \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(388\) |
| default | \(\frac {\left (c^{3} m^{3}+9 c^{3} m^{2}-3 c^{2} d \,m^{2}+26 c^{3} m -21 c^{2} d m +6 c \,d^{2} m +24 c^{3}-36 c^{2} d +24 d^{2} c -6 d^{3}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {d^{3} \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (4+m \right )}+\frac {\left (c^{3} m^{3}+3 c^{2} d \,m^{3}+9 c^{3} m^{2}+21 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+26 c^{3} m +36 c^{2} d m -24 c \,d^{2} m +6 d^{3} m +24 c^{3}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 c m +d m +12 c \right ) d^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+7 m +12\right )}+\frac {3 \left (c^{2} m^{2}+c d \,m^{2}+7 c^{2} m +4 c d m -d^{2} m +12 c^{2}\right ) d \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(388\) |
(a*(1+sin(f*x+e)))^m*(-3/2*(1+m)*((1/3*m^2+2/3*m+2)*d^2+c*m*(4+m)*d+c^2*(4 +m)*(3+m))*d*cos(2*f*x+2*e)+1/8*d^3*(3+m)*(2+m)*(1+m)*cos(4*f*x+4*e)-3/4*( 1+m)*(1/3*d*m+c*(4+m))*d^2*(2+m)*sin(3*f*x+3*e)+(3/4*m*(m^2+3*m+10)*d^3+9/ 4*c*(4+m)*(m^2+1/3*m+2)*d^2+3*c^2*m*(4+m)*(3+m)*d+c^3*(4+m)*(3+m)*(2+m))*s in(f*x+e)+3/8*(-1+m)*(m^2+3*m+10)*d^3+3/2*c*(4+m)*(m^2+m+4)*d^2+3/2*c^2*(- 1+m)*(4+m)*(3+m)*d+c^3*(4+m)*(3+m)*(2+m))/f/(m^4+10*m^3+35*m^2+50*m+24)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (133) = 266\).
Time = 0.29 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.03 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {{\left ({\left (d^{3} m^{3} + 6 \, d^{3} m^{2} + 11 \, d^{3} m + 6 \, d^{3}\right )} \cos \left (f x + e\right )^{4} + {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c^{3} + 24 \, c d^{2} + 3 \, {\left (3 \, c^{3} + 7 \, c^{2} d + 5 \, c d^{2} + d^{3}\right )} m^{2} - {\left ({\left (3 \, c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} m^{3} + 36 \, c^{2} d + 12 \, d^{3} + 3 \, {\left (8 \, c^{2} d + 5 \, c d^{2} + 3 \, d^{3}\right )} m^{2} + {\left (57 \, c^{2} d + 12 \, c d^{2} + 19 \, d^{3}\right )} m\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (13 \, c^{3} + 18 \, c^{2} d + 9 \, c d^{2} + 4 \, d^{3}\right )} m + {\left ({\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c^{3} + 24 \, c d^{2} + 3 \, {\left (3 \, c^{3} + 7 \, c^{2} d + 5 \, c d^{2} + d^{3}\right )} m^{2} - {\left ({\left (3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c d^{2} + 3 \, {\left (7 \, c d^{2} + d^{3}\right )} m^{2} + 2 \, {\left (21 \, c d^{2} + d^{3}\right )} m\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (13 \, c^{3} + 18 \, c^{2} d + 9 \, c d^{2} + 4 \, d^{3}\right )} m\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{4} + 10 \, f m^{3} + 35 \, f m^{2} + 50 \, f m + 24 \, f} \]
((d^3*m^3 + 6*d^3*m^2 + 11*d^3*m + 6*d^3)*cos(f*x + e)^4 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*m^3 + 24*c^3 + 24*c*d^2 + 3*(3*c^3 + 7*c^2*d + 5*c*d^2 + d^3)*m^2 - ((3*c^2*d + 3*c*d^2 + 2*d^3)*m^3 + 36*c^2*d + 12*d^3 + 3*(8*c^2 *d + 5*c*d^2 + 3*d^3)*m^2 + (57*c^2*d + 12*c*d^2 + 19*d^3)*m)*cos(f*x + e) ^2 + 2*(13*c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*m + ((c^3 + 3*c^2*d + 3*c*d^2 + d^3)*m^3 + 24*c^3 + 24*c*d^2 + 3*(3*c^3 + 7*c^2*d + 5*c*d^2 + d^3)*m^2 - ((3*c*d^2 + d^3)*m^3 + 24*c*d^2 + 3*(7*c*d^2 + d^3)*m^2 + 2*(21*c*d^2 + d^3)*m)*cos(f*x + e)^2 + 2*(13*c^3 + 18*c^2*d + 9*c*d^2 + 4*d^3)*m)*sin(f* x + e))*(a*sin(f*x + e) + a)^m/(f*m^4 + 10*f*m^3 + 35*f*m^2 + 50*f*m + 24* f)
Leaf count of result is larger than twice the leaf count of optimal. 4310 vs. \(2 (112) = 224\).
Time = 6.09 (sec) , antiderivative size = 4310, normalized size of antiderivative = 32.41 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
Piecewise((x*(c + d*sin(e))**3*(a*sin(e) + a)**m*cos(e), Eq(f, 0)), (-2*c* *3/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) - 9*c**2*d*sin(e + f*x)/(6*a**4*f*sin(e + f*x)**3 + 18 *a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) - 3*c**2*d/(6 *a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f* x) + 6*a**4*f) - 18*c*d**2*sin(e + f*x)**2/(6*a**4*f*sin(e + f*x)**3 + 18* a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) - 18*c*d**2*si n(e + f*x)/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4 *f*sin(e + f*x) + 6*a**4*f) - 6*c*d**2/(6*a**4*f*sin(e + f*x)**3 + 18*a**4 *f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) + 6*d**3*log(sin(e + f*x) + 1)*sin(e + f*x)**3/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) + 18*d**3*log(sin(e + f*x) + 1)*sin(e + f*x)**2/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) + 18*d**3*log(sin(e + f*x) + 1)*sin(e + f*x)/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f* sin(e + f*x) + 6*a**4*f) + 6*d**3*log(sin(e + f*x) + 1)/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) + 18*d**3*sin(e + f*x)**2/(6*a**4*f*sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x )**2 + 18*a**4*f*sin(e + f*x) + 6*a**4*f) + 27*d**3*sin(e + f*x)/(6*a**4*f *sin(e + f*x)**3 + 18*a**4*f*sin(e + f*x)**2 + 18*a**4*f*sin(e + f*x) +...
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (133) = 266\).
Time = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.21 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {\frac {3 \, {\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} c^{2} d {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} c d^{2} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} d^{3} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{3}}{a {\left (m + 1\right )}}}{f} \]
(3*(a^m*(m + 1)*sin(f*x + e)^2 + a^m*m*sin(f*x + e) - a^m)*c^2*d*(sin(f*x + e) + 1)^m/(m^2 + 3*m + 2) + 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)*c*d^2*(sin(f*x + e) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) + ((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)*d^3*(sin(f*x + e) + 1)^m/(m^4 + 10 *m^3 + 35*m^2 + 50*m + 24) + (a*sin(f*x + e) + a)^(m + 1)*c^3/(a*(m + 1))) /f
Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (133) = 266\).
Time = 0.35 (sec) , antiderivative size = 944, normalized size of antiderivative = 7.10 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
(3*((a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m^2 - 2*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m^2 + (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^2*m^2 + 3*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*m - 8*(a *sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a*m + 5*(a*sin(f*x + e) + a)*( a*sin(f*x + e) + a)^m*a^2*m + 2*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a )^m - 6*(a*sin(f*x + e) + a)^2*(a*sin(f*x + e) + a)^m*a + 6*(a*sin(f*x + e ) + a)*(a*sin(f*x + e) + a)^m*a^2)*c*d^2/(a^2*m^3 + 6*a^2*m^2 + 11*a^2*m + 6*a^2) + ((a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*m^3 - 3*(a*sin(f* x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m^3 + 3*(a*sin(f*x + e) + a)^2*(a*s in(f*x + e) + a)^m*a^2*m^3 - (a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a ^3*m^3 + 6*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*m^2 - 21*(a*sin(f *x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m^2 + 24*(a*sin(f*x + e) + a)^2*(a *sin(f*x + e) + a)^m*a^2*m^2 - 9*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a) ^m*a^3*m^2 + 11*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m*m - 42*(a*si n(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a*m + 57*(a*sin(f*x + e) + a)^2*( a*sin(f*x + e) + a)^m*a^2*m - 26*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a) ^m*a^3*m + 6*(a*sin(f*x + e) + a)^4*(a*sin(f*x + e) + a)^m - 24*(a*sin(f*x + e) + a)^3*(a*sin(f*x + e) + a)^m*a + 36*(a*sin(f*x + e) + a)^2*(a*sin(f *x + e) + a)^m*a^2 - 24*(a*sin(f*x + e) + a)*(a*sin(f*x + e) + a)^m*a^3)*d ^3/(a^3*m^4 + 10*a^3*m^3 + 35*a^3*m^2 + 50*a^3*m + 24*a^3) + (a*sin(f*x...
Time = 14.33 (sec) , antiderivative size = 703, normalized size of antiderivative = 5.29 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (192\,c\,d^2-144\,c^2\,d+208\,c^3\,m+21\,d^3\,m+192\,c^3\,\sin \left (e+f\,x\right )+192\,c^3-30\,d^3-24\,d^3\,\cos \left (2\,e+2\,f\,x\right )+6\,d^3\,\cos \left (4\,e+4\,f\,x\right )+72\,c^3\,m^2+8\,c^3\,m^3+6\,d^3\,m^2+3\,d^3\,m^3+208\,c^3\,m\,\sin \left (e+f\,x\right )+60\,d^3\,m\,\sin \left (e+f\,x\right )-144\,c^2\,d\,\cos \left (2\,e+2\,f\,x\right )+60\,c\,d^2\,m^2+72\,c^2\,d\,m^2+12\,c\,d^2\,m^3+12\,c^2\,d\,m^3-32\,d^3\,m\,\cos \left (2\,e+2\,f\,x\right )+11\,d^3\,m\,\cos \left (4\,e+4\,f\,x\right )-48\,c\,d^2\,\sin \left (3\,e+3\,f\,x\right )+72\,c^3\,m^2\,\sin \left (e+f\,x\right )+8\,c^3\,m^3\,\sin \left (e+f\,x\right )-4\,d^3\,m\,\sin \left (3\,e+3\,f\,x\right )+18\,d^3\,m^2\,\sin \left (e+f\,x\right )+6\,d^3\,m^3\,\sin \left (e+f\,x\right )-12\,d^3\,m^2\,\cos \left (2\,e+2\,f\,x\right )-4\,d^3\,m^3\,\cos \left (2\,e+2\,f\,x\right )+6\,d^3\,m^2\,\cos \left (4\,e+4\,f\,x\right )+d^3\,m^3\,\cos \left (4\,e+4\,f\,x\right )-6\,d^3\,m^2\,\sin \left (3\,e+3\,f\,x\right )-2\,d^3\,m^3\,\sin \left (3\,e+3\,f\,x\right )+96\,c\,d^2\,m+60\,c^2\,d\,m+144\,c\,d^2\,\sin \left (e+f\,x\right )-60\,c\,d^2\,m^2\,\cos \left (2\,e+2\,f\,x\right )-96\,c^2\,d\,m^2\,\cos \left (2\,e+2\,f\,x\right )-12\,c\,d^2\,m^3\,\cos \left (2\,e+2\,f\,x\right )-12\,c^2\,d\,m^3\,\cos \left (2\,e+2\,f\,x\right )-42\,c\,d^2\,m^2\,\sin \left (3\,e+3\,f\,x\right )-6\,c\,d^2\,m^3\,\sin \left (3\,e+3\,f\,x\right )+60\,c\,d^2\,m\,\sin \left (e+f\,x\right )+288\,c^2\,d\,m\,\sin \left (e+f\,x\right )-48\,c\,d^2\,m\,\cos \left (2\,e+2\,f\,x\right )-228\,c^2\,d\,m\,\cos \left (2\,e+2\,f\,x\right )-84\,c\,d^2\,m\,\sin \left (3\,e+3\,f\,x\right )+78\,c\,d^2\,m^2\,\sin \left (e+f\,x\right )+168\,c^2\,d\,m^2\,\sin \left (e+f\,x\right )+18\,c\,d^2\,m^3\,\sin \left (e+f\,x\right )+24\,c^2\,d\,m^3\,\sin \left (e+f\,x\right )\right )}{8\,f\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
((a*(sin(e + f*x) + 1))^m*(192*c*d^2 - 144*c^2*d + 208*c^3*m + 21*d^3*m + 192*c^3*sin(e + f*x) + 192*c^3 - 30*d^3 - 24*d^3*cos(2*e + 2*f*x) + 6*d^3* cos(4*e + 4*f*x) + 72*c^3*m^2 + 8*c^3*m^3 + 6*d^3*m^2 + 3*d^3*m^3 + 208*c^ 3*m*sin(e + f*x) + 60*d^3*m*sin(e + f*x) - 144*c^2*d*cos(2*e + 2*f*x) + 60 *c*d^2*m^2 + 72*c^2*d*m^2 + 12*c*d^2*m^3 + 12*c^2*d*m^3 - 32*d^3*m*cos(2*e + 2*f*x) + 11*d^3*m*cos(4*e + 4*f*x) - 48*c*d^2*sin(3*e + 3*f*x) + 72*c^3 *m^2*sin(e + f*x) + 8*c^3*m^3*sin(e + f*x) - 4*d^3*m*sin(3*e + 3*f*x) + 18 *d^3*m^2*sin(e + f*x) + 6*d^3*m^3*sin(e + f*x) - 12*d^3*m^2*cos(2*e + 2*f* x) - 4*d^3*m^3*cos(2*e + 2*f*x) + 6*d^3*m^2*cos(4*e + 4*f*x) + d^3*m^3*cos (4*e + 4*f*x) - 6*d^3*m^2*sin(3*e + 3*f*x) - 2*d^3*m^3*sin(3*e + 3*f*x) + 96*c*d^2*m + 60*c^2*d*m + 144*c*d^2*sin(e + f*x) - 60*c*d^2*m^2*cos(2*e + 2*f*x) - 96*c^2*d*m^2*cos(2*e + 2*f*x) - 12*c*d^2*m^3*cos(2*e + 2*f*x) - 1 2*c^2*d*m^3*cos(2*e + 2*f*x) - 42*c*d^2*m^2*sin(3*e + 3*f*x) - 6*c*d^2*m^3 *sin(3*e + 3*f*x) + 60*c*d^2*m*sin(e + f*x) + 288*c^2*d*m*sin(e + f*x) - 4 8*c*d^2*m*cos(2*e + 2*f*x) - 228*c^2*d*m*cos(2*e + 2*f*x) - 84*c*d^2*m*sin (3*e + 3*f*x) + 78*c*d^2*m^2*sin(e + f*x) + 168*c^2*d*m^2*sin(e + f*x) + 1 8*c*d^2*m^3*sin(e + f*x) + 24*c^2*d*m^3*sin(e + f*x)))/(8*f*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))