Integrand size = 31, antiderivative size = 133 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {(c-d)^3 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac {3 (c-d)^2 d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac {3 (c-d) d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}+\frac {d^3 (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)} \] Output:
(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 = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {(a (1+\sin (e+f x)))^{1+m} \left (\frac {a^3 (c-d)^3}{1+m}+\frac {3 a^3 (c-d)^2 d (1+\sin (e+f x))}{2+m}+\frac {3 a^3 (c-d) d^2 (1+\sin (e+f x))^2}{3+m}+\frac {d^3 (a+a \sin (e+f x))^3}{4+m}\right )}{a^4 f} \] Input:
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3,x]
Output:
((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 = 0.37 (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 |
| \(\displaystyle \frac {\frac {a^3 (c-d)^3 (a \sin (e+f x)+a)^{m+1}}{m+1}+\frac {3 a^2 d (c-d)^2 (a \sin (e+f x)+a)^{m+2}}{m+2}+\frac {3 a d^2 (c-d) (a \sin (e+f x)+a)^{m+3}}{m+3}+\frac {d^3 (a \sin (e+f x)+a)^{m+4}}{m+4}}{a^4 f}\) |
Input:
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3,x]
Output:
((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 )
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 = 3.97 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.97
| method | result | size |
| parallelrisch | \(\frac {\left (-\frac {3 \left (1+m \right ) d \left (\left (\frac {1}{3} m^{2}+\frac {2}{3} m +2\right ) d^{2}+c m \left (4+m \right ) d +c^{2} \left (4+m \right ) \left (3+m \right )\right ) \cos \left (2 f x +2 e \right )}{2}+\frac {d^{3} \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \cos \left (4 f x +4 e \right )}{8}-\frac {3 \left (1+m \right ) \left (\frac {d m}{3}+c \left (4+m \right )\right ) \left (2+m \right ) d^{2} \sin \left (3 f x +3 e \right )}{4}+\left (\frac {3 m \left (m^{2}+3 m +10\right ) d^{3}}{4}+\frac {9 \left (4+m \right ) c \left (m^{2}+\frac {1}{3} m +2\right ) d^{2}}{4}+3 c^{2} m \left (4+m \right ) \left (3+m \right ) d +c^{3} \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right ) \sin \left (f x +e \right )+\frac {3 \left (-1+m \right ) \left (m^{2}+3 m +10\right ) d^{3}}{8}+\frac {3 c \left (4+m \right ) \left (m^{2}+m +4\right ) d^{2}}{2}+\frac {3 c^{2} \left (-1+m \right ) \left (4+m \right ) \left (3+m \right ) d}{2}+c^{3} \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(262\) |
| derivativedivides | \(\frac {\left (c^{3} m^{3}+9 c^{3} m^{2}-3 c^{2} d \,m^{2}+26 c^{3} m -21 c^{2} d m +6 c \,d^{2} m +24 c^{3}-36 c^{2} d +24 c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {d^{3} \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (4+m \right )}+\frac {\left (c^{3} m^{3}+3 c^{2} d \,m^{3}+9 c^{3} m^{2}+21 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+26 c^{3} m +36 c^{2} d m -24 c \,d^{2} m +6 d^{3} m +24 c^{3}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 c m +d m +12 c \right ) d^{2} \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+7 m +12\right )}+\frac {3 \left (c^{2} m^{2}+c d \,m^{2}+7 c^{2} m +4 c d m -d^{2} m +12 c^{2}\right ) d \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(388\) |
| default | \(\frac {\left (c^{3} m^{3}+9 c^{3} m^{2}-3 c^{2} d \,m^{2}+26 c^{3} m -21 c^{2} d m +6 c \,d^{2} m +24 c^{3}-36 c^{2} d +24 c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {d^{3} \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (4+m \right )}+\frac {\left (c^{3} m^{3}+3 c^{2} d \,m^{3}+9 c^{3} m^{2}+21 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+26 c^{3} m +36 c^{2} d m -24 c \,d^{2} m +6 d^{3} m +24 c^{3}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (3 c m +d m +12 c \right ) d^{2} \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+7 m +12\right )}+\frac {3 \left (c^{2} m^{2}+c d \,m^{2}+7 c^{2} m +4 c d m -d^{2} m +12 c^{2}\right ) d \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+9 m^{2}+26 m +24\right )}\) | \(388\) |
Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBO SE)
Output:
(-3/2*(1+m)*d*((1/3*m^2+2/3*m+2)*d^2+c*m*(4+m)*d+c^2*(4+m)*(3+m))*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) )*(2+m)*d^2*sin(3*f*x+3*e)+(3/4*m*(m^2+3*m+10)*d^3+9/4*(4+m)*c*(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))*sin(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))*(a*(1+sin(f*x+e)))^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 = 0.11 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.03 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {{\left ({\left (d^{3} m^{3} + 6 \, d^{3} m^{2} + 11 \, d^{3} m + 6 \, d^{3}\right )} \cos \left (f x + e\right )^{4} + {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c^{3} + 24 \, c d^{2} + 3 \, {\left (3 \, c^{3} + 7 \, c^{2} d + 5 \, c d^{2} + d^{3}\right )} m^{2} - {\left ({\left (3 \, c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} m^{3} + 36 \, c^{2} d + 12 \, d^{3} + 3 \, {\left (8 \, c^{2} d + 5 \, c d^{2} + 3 \, d^{3}\right )} m^{2} + {\left (57 \, c^{2} d + 12 \, c d^{2} + 19 \, d^{3}\right )} m\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (13 \, c^{3} + 18 \, c^{2} d + 9 \, c d^{2} + 4 \, d^{3}\right )} m + {\left ({\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c^{3} + 24 \, c d^{2} + 3 \, {\left (3 \, c^{3} + 7 \, c^{2} d + 5 \, c d^{2} + d^{3}\right )} m^{2} - {\left ({\left (3 \, c d^{2} + d^{3}\right )} m^{3} + 24 \, c d^{2} + 3 \, {\left (7 \, c d^{2} + d^{3}\right )} m^{2} + 2 \, {\left (21 \, c d^{2} + d^{3}\right )} m\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (13 \, c^{3} + 18 \, c^{2} d + 9 \, c d^{2} + 4 \, d^{3}\right )} m\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{4} + 10 \, f m^{3} + 35 \, f m^{2} + 50 \, f m + 24 \, f} \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x, algorithm="f ricas")
Output:
((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 = 6.50 (sec) , antiderivative size = 4310, normalized size of antiderivative = 32.41 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**3,x)
Output:
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 = 0.04 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.21 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {\frac {3 \, {\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} c^{2} d {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} c d^{2} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} d^{3} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{3}}{a {\left (m + 1\right )}}}{f} \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x, algorithm="m axima")
Output:
(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. 571 vs. \(2 (133) = 266\).
Time = 0.13 (sec) , antiderivative size = 571, normalized size of antiderivative = 4.29 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x, algorithm="g iac")
Output:
(3*((a*sin(f*x + e) + a)^m*m*sin(f*x + e)^2 + (a*sin(f*x + e) + a)^m*m*sin (f*x + e) + (a*sin(f*x + e) + a)^m*sin(f*x + e)^2 - (a*sin(f*x + e) + a)^m )*c^2*d/(m^2 + 3*m + 2) + 3*((a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^3 + ( a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^2 + 3*(a*sin(f*x + e) + a)^m*m*sin( f*x + e)^3 + (a*sin(f*x + e) + a)^m*m*sin(f*x + e)^2 + 2*(a*sin(f*x + e) + a)^m*sin(f*x + e)^3 - 2*(a*sin(f*x + e) + a)^m*m*sin(f*x + e) + 2*(a*sin( f*x + e) + a)^m)*c*d^2/(m^3 + 6*m^2 + 11*m + 6) + ((a*sin(f*x + e) + a)^m* m^3*sin(f*x + e)^4 + (a*sin(f*x + e) + a)^m*m^3*sin(f*x + e)^3 + 6*(a*sin( f*x + e) + a)^m*m^2*sin(f*x + e)^4 + 3*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^3 + 11*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^4 - 3*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^2 + 2*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^3 + 6*(a *sin(f*x + e) + a)^m*sin(f*x + e)^4 - 3*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^2 + 6*(a*sin(f*x + e) + a)^m*m*sin(f*x + e) - 6*(a*sin(f*x + e) + a)^m )*d^3/(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
Time = 36.70 (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 )} \] Input:
int(cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^3,x)
Output:
((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))
Time = 0.17 (sec) , antiderivative size = 544, normalized size of antiderivative = 4.09 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx=\frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (24 \sin \left (f x +e \right )^{3} c \,d^{2}+6 \sin \left (f x +e \right )^{4} d^{3}-6 d^{3}+24 \sin \left (f x +e \right )^{2} c^{2} d \,m^{2}+57 \sin \left (f x +e \right )^{2} c^{2} d m +3 \sin \left (f x +e \right )^{2} c \,d^{2} m^{3}+15 \sin \left (f x +e \right )^{2} c \,d^{2} m^{2}+12 \sin \left (f x +e \right )^{2} c \,d^{2} m +3 \sin \left (f x +e \right ) c^{2} d \,m^{3}+21 \sin \left (f x +e \right ) c^{2} d \,m^{2}+36 \sin \left (f x +e \right ) c^{2} d m -6 \sin \left (f x +e \right ) c \,d^{2} m^{2}-24 \sin \left (f x +e \right ) c \,d^{2} m +\sin \left (f x +e \right ) c^{3} m^{3}+3 \sin \left (f x +e \right )^{3} c \,d^{2} m^{3}+21 \sin \left (f x +e \right )^{3} c \,d^{2} m^{2}+42 \sin \left (f x +e \right )^{3} c \,d^{2} m +3 \sin \left (f x +e \right )^{2} c^{2} d \,m^{3}+\sin \left (f x +e \right )^{4} d^{3} m^{3}+\sin \left (f x +e \right )^{3} d^{3} m^{3}+24 c^{3}+6 \sin \left (f x +e \right )^{4} d^{3} m^{2}+11 \sin \left (f x +e \right )^{4} d^{3} m +3 \sin \left (f x +e \right )^{3} d^{3} m^{2}+2 \sin \left (f x +e \right )^{3} d^{3} m -3 \sin \left (f x +e \right )^{2} d^{3} m^{2}-3 \sin \left (f x +e \right )^{2} d^{3} m +9 \sin \left (f x +e \right ) c^{3} m^{2}+c^{3} m^{3}+9 c^{3} m^{2}+26 c^{3} m +24 \sin \left (f x +e \right ) c^{3}+26 \sin \left (f x +e \right ) c^{3} m +6 \sin \left (f x +e \right ) d^{3} m -3 c^{2} d \,m^{2}-21 c^{2} d m +6 c \,d^{2} m +36 \sin \left (f x +e \right )^{2} c^{2} d -36 c^{2} d +24 c \,d^{2}\right )}{f \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \] Input:
int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x)
Output:
((sin(e + f*x)*a + a)**m*(sin(e + f*x)**4*d**3*m**3 + 6*sin(e + f*x)**4*d* *3*m**2 + 11*sin(e + f*x)**4*d**3*m + 6*sin(e + f*x)**4*d**3 + 3*sin(e + f *x)**3*c*d**2*m**3 + 21*sin(e + f*x)**3*c*d**2*m**2 + 42*sin(e + f*x)**3*c *d**2*m + 24*sin(e + f*x)**3*c*d**2 + sin(e + f*x)**3*d**3*m**3 + 3*sin(e + f*x)**3*d**3*m**2 + 2*sin(e + f*x)**3*d**3*m + 3*sin(e + f*x)**2*c**2*d* m**3 + 24*sin(e + f*x)**2*c**2*d*m**2 + 57*sin(e + f*x)**2*c**2*d*m + 36*s in(e + f*x)**2*c**2*d + 3*sin(e + f*x)**2*c*d**2*m**3 + 15*sin(e + f*x)**2 *c*d**2*m**2 + 12*sin(e + f*x)**2*c*d**2*m - 3*sin(e + f*x)**2*d**3*m**2 - 3*sin(e + f*x)**2*d**3*m + sin(e + f*x)*c**3*m**3 + 9*sin(e + f*x)*c**3*m **2 + 26*sin(e + f*x)*c**3*m + 24*sin(e + f*x)*c**3 + 3*sin(e + f*x)*c**2* d*m**3 + 21*sin(e + f*x)*c**2*d*m**2 + 36*sin(e + f*x)*c**2*d*m - 6*sin(e + f*x)*c*d**2*m**2 - 24*sin(e + f*x)*c*d**2*m + 6*sin(e + f*x)*d**3*m + c* *3*m**3 + 9*c**3*m**2 + 26*c**3*m + 24*c**3 - 3*c**2*d*m**2 - 21*c**2*d*m - 36*c**2*d + 6*c*d**2*m + 24*c*d**2 - 6*d**3))/(f*(m**4 + 10*m**3 + 35*m* *2 + 50*m + 24))