\(\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx\) [917]

Optimal result

Integrand size = 31, antiderivative size = 170 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\frac {(c-d)^4 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac {4 (c-d)^3 d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac {6 (c-d)^2 d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}+\frac {4 (c-d) d^3 (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac {d^4 (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)} \] Output:

(c-d)^4*(a+a*sin(f*x+e))^(1+m)/a/f/(1+m)+4*(c-d)^3*d*(a+a*sin(f*x+e))^(2+m 
)/a^2/f/(2+m)+6*(c-d)^2*d^2*(a+a*sin(f*x+e))^(3+m)/a^3/f/(3+m)+4*(c-d)*d^3 
*(a+a*sin(f*x+e))^(4+m)/a^4/f/(4+m)+d^4*(a+a*sin(f*x+e))^(5+m)/a^5/f/(5+m)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\frac {(a (1+\sin (e+f x)))^{1+m} \left (\frac {a^4 (c-d)^4}{1+m}+\frac {4 a^4 (c-d)^3 d (1+\sin (e+f x))}{2+m}+\frac {6 a^4 (c-d)^2 d^2 (1+\sin (e+f x))^2}{3+m}+\frac {4 a^4 (c-d) d^3 (1+\sin (e+f x))^3}{4+m}+\frac {d^4 (a+a \sin (e+f x))^4}{5+m}\right )}{a^5 f} \] Input:

Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]
 

Output:

((a*(1 + Sin[e + f*x]))^(1 + m)*((a^4*(c - d)^4)/(1 + m) + (4*a^4*(c - d)^ 
3*d*(1 + Sin[e + f*x]))/(2 + m) + (6*a^4*(c - d)^2*d^2*(1 + Sin[e + f*x])^ 
2)/(3 + m) + (4*a^4*(c - d)*d^3*(1 + Sin[e + f*x])^3)/(4 + m) + (d^4*(a + 
a*Sin[e + f*x])^4)/(5 + m)))/(a^5*f)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 157, 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.

\(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^4dx\)

\(\Big \downarrow \) 3312

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m (a c+a d \sin (e+f x))^4}{a^4}d(a \sin (e+f x))}{a f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (\sin (e+f x) a+a)^m (a c+a d \sin (e+f x))^4d(a \sin (e+f x))}{a^5 f}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\int \left (a^4 (c-d)^4 (\sin (e+f x) a+a)^m+4 a^3 (c-d)^3 d (\sin (e+f x) a+a)^{m+1}+6 a^2 (c-d)^2 d^2 (\sin (e+f x) a+a)^{m+2}+4 a (c-d) d^3 (\sin (e+f x) a+a)^{m+3}+d^4 (\sin (e+f x) a+a)^{m+4}\right )d(a \sin (e+f x))}{a^5 f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^4 (c-d)^4 (a \sin (e+f x)+a)^{m+1}}{m+1}+\frac {4 a^3 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{m+2}+\frac {6 a^2 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{m+3}+\frac {4 a d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{m+4}+\frac {d^4 (a \sin (e+f x)+a)^{m+5}}{m+5}}{a^5 f}\)

Input:

Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]
 

Output:

((a^4*(c - d)^4*(a + a*Sin[e + f*x])^(1 + m))/(1 + m) + (4*a^3*(c - d)^3*d 
*(a + a*Sin[e + f*x])^(2 + m))/(2 + m) + (6*a^2*(c - d)^2*d^2*(a + a*Sin[e 
 + f*x])^(3 + m))/(3 + m) + (4*a*(c - d)*d^3*(a + a*Sin[e + f*x])^(4 + m)) 
/(4 + m) + (d^4*(a + a*Sin[e + f*x])^(5 + m))/(5 + m))/(a^5*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 53

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])
 

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 3312

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(428\) vs. \(2(170)=340\).

Time = 8.93 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.52

method	result	size
parallelrisch	\(\frac {\left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m} \left (-2 \left (1+m \right ) \left (\frac {m \left (m^{2}+5 m +18\right ) d^{3}}{4}+c \left (5+m \right ) \left (m^{2}+2 m +6\right ) d^{2}+\frac {3 c^{2} m \left (5+m \right ) \left (4+m \right ) d}{2}+c^{3} \left (5+m \right ) \left (4+m \right ) \left (3+m \right )\right ) d \cos \left (2 f x +2 e \right )+\frac {\left (1+m \right ) \left (3+m \right ) \left (\frac {d m}{4}+c \left (5+m \right )\right ) \left (2+m \right ) d^{3} \cos \left (4 f x +4 e \right )}{2}-\frac {3 \left (1+m \right ) \left (\left (\frac {5}{24} m^{2}+\frac {19}{24} m +\frac {5}{2}\right ) d^{2}+\frac {2 c m \left (5+m \right ) d}{3}+c^{2} \left (5+m \right ) \left (4+m \right )\right ) \left (2+m \right ) d^{2} \sin \left (3 f x +3 e \right )}{2}+\frac {d^{4} \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \sin \left (5 f x +5 e \right )}{16}+\left (\left (\frac {5}{8} m^{4}+\frac {13}{4} m^{3}+\frac {103}{8} m^{2}+\frac {5}{4} m +15\right ) d^{4}+3 c m \left (5+m \right ) \left (m^{2}+3 m +10\right ) d^{3}+\frac {9 \left (4+m \right ) c^{2} \left (5+m \right ) \left (m^{2}+\frac {1}{3} m +2\right ) d^{2}}{2}+4 m \,c^{3} \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) d +c^{4} \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right ) \sin \left (f x +e \right )+\left (24+\frac {3}{8} m^{4}+\frac {9}{4} m^{3}+\frac {81}{8} m^{2}+\frac {33}{4} m \right ) d^{4}+\frac {3 c \left (-1+m \right ) \left (5+m \right ) \left (m^{2}+3 m +10\right ) d^{3}}{2}+3 c^{2} \left (5+m \right ) \left (4+m \right ) \left (m^{2}+m +4\right ) d^{2}+2 c^{3} \left (-1+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) d +c^{4} \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right )\right )}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) f}\)	\(429\)
derivativedivides	\(\frac {\left (c^{4} m^{4}+14 c^{4} m^{3}-4 c^{3} d \,m^{3}+71 c^{4} m^{2}-48 c^{3} d \,m^{2}+12 c^{2} d^{2} m^{2}+154 c^{4} m -188 c^{3} d m +108 c^{2} d^{2} m -24 c \,d^{3} m +120 c^{4}-240 c^{3} d +240 c^{2} d^{2}-120 c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {d^{4} \sin \left (f x +e \right )^{5} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (5+m \right )}+\frac {\left (c^{4} m^{4}+4 c^{3} d \,m^{4}+14 c^{4} m^{3}+48 c^{3} d \,m^{3}-12 c^{2} d^{2} m^{3}+71 c^{4} m^{2}+188 c^{3} d \,m^{2}-108 c^{2} d^{2} m^{2}+24 c \,d^{3} m^{2}+154 c^{4} m +240 c^{3} d m -240 c^{2} d^{2} m +120 c \,d^{3} m -24 d^{4} m +120 c^{4}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {d^{3} \left (4 c m +d m +20 c \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+9 m +20\right )}+\frac {2 \left (3 c^{2} m^{2}+2 c d \,m^{2}+27 c^{2} m +10 c d m -2 d^{2} m +60 c^{2}\right ) d^{2} \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 \left (2 c^{3} m^{3}+3 c^{2} d \,m^{3}+24 c^{3} m^{2}+27 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+94 c^{3} m +60 c^{2} d m -30 c \,d^{2} m +6 d^{3} m +120 c^{3}\right ) d \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}\)	\(618\)
default	\(\frac {\left (c^{4} m^{4}+14 c^{4} m^{3}-4 c^{3} d \,m^{3}+71 c^{4} m^{2}-48 c^{3} d \,m^{2}+12 c^{2} d^{2} m^{2}+154 c^{4} m -188 c^{3} d m +108 c^{2} d^{2} m -24 c \,d^{3} m +120 c^{4}-240 c^{3} d +240 c^{2} d^{2}-120 c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {d^{4} \sin \left (f x +e \right )^{5} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (5+m \right )}+\frac {\left (c^{4} m^{4}+4 c^{3} d \,m^{4}+14 c^{4} m^{3}+48 c^{3} d \,m^{3}-12 c^{2} d^{2} m^{3}+71 c^{4} m^{2}+188 c^{3} d \,m^{2}-108 c^{2} d^{2} m^{2}+24 c \,d^{3} m^{2}+154 c^{4} m +240 c^{3} d m -240 c^{2} d^{2} m +120 c \,d^{3} m -24 d^{4} m +120 c^{4}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {d^{3} \left (4 c m +d m +20 c \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+9 m +20\right )}+\frac {2 \left (3 c^{2} m^{2}+2 c d \,m^{2}+27 c^{2} m +10 c d m -2 d^{2} m +60 c^{2}\right ) d^{2} \sin \left (f x +e \right )^{3} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 \left (2 c^{3} m^{3}+3 c^{2} d \,m^{3}+24 c^{3} m^{2}+27 c^{2} d \,m^{2}-6 c \,d^{2} m^{2}+94 c^{3} m +60 c^{2} d m -30 c \,d^{2} m +6 d^{3} m +120 c^{3}\right ) d \sin \left (f x +e \right )^{2} {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}\)	\(618\)

Input:

int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x,method=_RETURNVERBO 
SE)
 

Output:

(a*(1+sin(f*x+e)))^m*(-2*(1+m)*(1/4*m*(m^2+5*m+18)*d^3+c*(5+m)*(m^2+2*m+6) 
*d^2+3/2*c^2*m*(5+m)*(4+m)*d+c^3*(5+m)*(4+m)*(3+m))*d*cos(2*f*x+2*e)+1/2*( 
1+m)*(3+m)*(1/4*d*m+c*(5+m))*(2+m)*d^3*cos(4*f*x+4*e)-3/2*(1+m)*((5/24*m^2 
+19/24*m+5/2)*d^2+2/3*c*m*(5+m)*d+c^2*(5+m)*(4+m))*(2+m)*d^2*sin(3*f*x+3*e 
)+1/16*d^4*(4+m)*(3+m)*(2+m)*(1+m)*sin(5*f*x+5*e)+((5/8*m^4+13/4*m^3+103/8 
*m^2+5/4*m+15)*d^4+3*c*m*(5+m)*(m^2+3*m+10)*d^3+9/2*(4+m)*c^2*(5+m)*(m^2+1 
/3*m+2)*d^2+4*m*c^3*(5+m)*(4+m)*(3+m)*d+c^4*(5+m)*(4+m)*(3+m)*(2+m))*sin(f 
*x+e)+(24+3/8*m^4+9/4*m^3+81/8*m^2+33/4*m)*d^4+3/2*c*(-1+m)*(5+m)*(m^2+3*m 
+10)*d^3+3*c^2*(5+m)*(4+m)*(m^2+m+4)*d^2+2*c^3*(-1+m)*(5+m)*(4+m)*(3+m)*d+ 
c^4*(5+m)*(4+m)*(3+m)*(2+m))/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (170) = 340\).

Time = 0.17 (sec) , antiderivative size = 744, normalized size of antiderivative = 4.38 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx =\text {Too large to display} \] Input:

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="f 
ricas")
 

Output:

((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*m^4 + ((4*c*d^3 + d^4)*m^4 + 
120*c*d^3 + 2*(22*c*d^3 + 3*d^4)*m^3 + (164*c*d^3 + 11*d^4)*m^2 + 2*(122*c 
*d^3 + 3*d^4)*m)*cos(f*x + e)^4 + 120*c^4 + 240*c^2*d^2 + 24*d^4 + 2*(7*c^ 
4 + 24*c^3*d + 30*c^2*d^2 + 16*c*d^3 + 3*d^4)*m^3 + (71*c^4 + 188*c^3*d + 
186*c^2*d^2 + 92*c*d^3 + 23*d^4)*m^2 - 2*((2*c^3*d + 3*c^2*d^2 + 4*c*d^3 + 
 d^4)*m^4 + 120*c^3*d + 120*c*d^3 + 2*(13*c^3*d + 15*c^2*d^2 + 19*c*d^3 + 
3*d^4)*m^3 + (118*c^3*d + 87*c^2*d^2 + 128*c*d^3 + 17*d^4)*m^2 + 2*(107*c^ 
3*d + 30*c^2*d^2 + 107*c*d^3 + 6*d^4)*m)*cos(f*x + e)^2 + 2*(77*c^4 + 120* 
c^3*d + 114*c^2*d^2 + 80*c*d^3 + 9*d^4)*m + ((c^4 + 4*c^3*d + 6*c^2*d^2 + 
4*c*d^3 + d^4)*m^4 + (d^4*m^4 + 10*d^4*m^3 + 35*d^4*m^2 + 50*d^4*m + 24*d^ 
4)*cos(f*x + e)^4 + 120*c^4 + 240*c^2*d^2 + 24*d^4 + 2*(7*c^4 + 24*c^3*d + 
 30*c^2*d^2 + 16*c*d^3 + 3*d^4)*m^3 + (71*c^4 + 188*c^3*d + 186*c^2*d^2 + 
92*c*d^3 + 23*d^4)*m^2 - 2*((3*c^2*d^2 + 2*c*d^3 + d^4)*m^4 + 120*c^2*d^2 
+ 24*d^4 + 4*(9*c^2*d^2 + 4*c*d^3 + 2*d^4)*m^3 + (147*c^2*d^2 + 34*c*d^3 + 
 29*d^4)*m^2 + 2*(117*c^2*d^2 + 10*c*d^3 + 23*d^4)*m)*cos(f*x + e)^2 + 2*( 
77*c^4 + 120*c^3*d + 114*c^2*d^2 + 80*c*d^3 + 9*d^4)*m)*sin(f*x + e))*(a*s 
in(f*x + e) + a)^m/(f*m^5 + 15*f*m^4 + 85*f*m^3 + 225*f*m^2 + 274*f*m + 12 
0*f)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9238 vs. \(2 (144) = 288\).

Time = 15.16 (sec) , antiderivative size = 9238, normalized size of antiderivative = 54.34 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\text {Too large to display} \] Input:

integrate(cos(f*x+e)*(a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**4,x)
 

Output:

Piecewise((x*(c + d*sin(e))**4*(a*sin(e) + a)**m*cos(e), Eq(f, 0)), (-3*c* 
*4/(12*a**5*f*sin(e + f*x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin( 
e + f*x)**2 + 48*a**5*f*sin(e + f*x) + 12*a**5*f) - 16*c**3*d*sin(e + f*x) 
/(12*a**5*f*sin(e + f*x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e 
+ f*x)**2 + 48*a**5*f*sin(e + f*x) + 12*a**5*f) - 4*c**3*d/(12*a**5*f*sin( 
e + f*x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a 
**5*f*sin(e + f*x) + 12*a**5*f) - 36*c**2*d**2*sin(e + f*x)**2/(12*a**5*f* 
sin(e + f*x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 
48*a**5*f*sin(e + f*x) + 12*a**5*f) - 24*c**2*d**2*sin(e + f*x)/(12*a**5*f 
*sin(e + f*x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 
 48*a**5*f*sin(e + f*x) + 12*a**5*f) - 6*c**2*d**2/(12*a**5*f*sin(e + f*x) 
**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a**5*f*si 
n(e + f*x) + 12*a**5*f) - 48*c*d**3*sin(e + f*x)**3/(12*a**5*f*sin(e + f*x 
)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a**5*f*s 
in(e + f*x) + 12*a**5*f) - 72*c*d**3*sin(e + f*x)**2/(12*a**5*f*sin(e + f* 
x)**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a**5*f* 
sin(e + f*x) + 12*a**5*f) - 48*c*d**3*sin(e + f*x)/(12*a**5*f*sin(e + f*x) 
**4 + 48*a**5*f*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a**5*f*si 
n(e + f*x) + 12*a**5*f) - 12*c*d**3/(12*a**5*f*sin(e + f*x)**4 + 48*a**5*f 
*sin(e + f*x)**3 + 72*a**5*f*sin(e + f*x)**2 + 48*a**5*f*sin(e + f*x) +...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (170) = 340\).

Time = 0.05 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.69 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\frac {\frac {4 \, {\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} c^{3} d {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} c^{2} d^{2} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} c d^{3} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (f x + e\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 24 \, a^{m} m \sin \left (f x + e\right ) + 24 \, a^{m}\right )} d^{4} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{4}}{a {\left (m + 1\right )}}}{f} \] Input:

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="m 
axima")
 

Output:

(4*(a^m*(m + 1)*sin(f*x + e)^2 + a^m*m*sin(f*x + e) - a^m)*c^3*d*(sin(f*x 
+ e) + 1)^m/(m^2 + 3*m + 2) + 6*((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^2*d^2*(sin(f*x 
+ e) + 1)^m/(m^3 + 6*m^2 + 11*m + 6) + 4*((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)*c*d^3*(sin(f*x + e) + 1)^m/(m^ 
4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((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)*d^4*(sin(f*x + e) + 1)^m/(m^5 + 15*m^4 + 85* 
m^3 + 225*m^2 + 274*m + 120) + (a*sin(f*x + e) + a)^(m + 1)*c^4/(a*(m + 1) 
))/f
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (170) = 340\).

Time = 0.14 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.75 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\text {Too large to display} \] Input:

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x, algorithm="g 
iac")
 

Output:

(4*((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^3*d/(m^2 + 3*m + 2) + 6*((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^2*d^2/(m^3 + 6*m^2 + 11*m + 6) + 4*((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)*c*d^3/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((a*sin(f*x + e) + a)^m* 
m^4*sin(f*x + e)^5 + (a*sin(f*x + e) + a)^m*m^4*sin(f*x + e)^4 + 10*(a*sin 
(f*x + e) + a)^m*m^3*sin(f*x + e)^5 + 6*(a*sin(f*x + e) + a)^m*m^3*sin(f*x 
 + e)^4 + 35*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^5 - 4*(a*sin(f*x + e) 
 + a)^m*m^3*sin(f*x + e)^3 + 11*(a*sin(f*x + e) + a)^m*m^2*sin(f*x + e)^4 
+ 50*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^5 - 12*(a*sin(f*x + e) + a)^m*m 
^2*sin(f*x + e)^3 + 6*(a*sin(f*x + e) + a)^m*m*sin(f*x + e)^4 + 24*(a*s...
 

Mupad [B] (verification not implemented)

Time = 40.59 (sec) , antiderivative size = 1656, normalized size of antiderivative = 9.74 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx=\text {Too large to display} \] Input:

int(cos(e + f*x)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^4,x)
 

Output:

exp(- e*5i - f*x*5i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x 
*1i)*1i)/2))^m*((exp(e*6i + f*x*6i)*(2464*c^4*m + 20*d^4*m + 1920*c^4 + 24 
0*d^4 + 2880*c^2*d^2 + 1136*c^4*m^2 + 224*c^4*m^3 + 16*c^4*m^4 + 206*d^4*m 
^2 + 52*d^4*m^3 + 10*d^4*m^4 + 1776*c^2*d^2*m + 1200*c*d^3*m^2 + 3008*c^3* 
d*m^2 + 384*c*d^3*m^3 + 768*c^3*d*m^3 + 48*c*d^3*m^4 + 64*c^3*d*m^4 + 1800 
*c^2*d^2*m^2 + 672*c^2*d^2*m^3 + 72*c^2*d^2*m^4 + 2400*c*d^3*m + 3840*c^3* 
d*m))/(32*f*(m*274i + m^2*225i + m^3*85i + m^4*15i + m^5*1i + 120i)) - (ex 
p(e*4i + f*x*4i)*(2464*c^4*m + 20*d^4*m + 1920*c^4 + 240*d^4 + 2880*c^2*d^ 
2 + 1136*c^4*m^2 + 224*c^4*m^3 + 16*c^4*m^4 + 206*d^4*m^2 + 52*d^4*m^3 + 1 
0*d^4*m^4 + 1776*c^2*d^2*m + 1200*c*d^3*m^2 + 3008*c^3*d*m^2 + 384*c*d^3*m 
^3 + 768*c^3*d*m^3 + 48*c*d^3*m^4 + 64*c^3*d*m^4 + 1800*c^2*d^2*m^2 + 672* 
c^2*d^2*m^3 + 72*c^2*d^2*m^4 + 2400*c*d^3*m + 3840*c^3*d*m))/(32*f*(m*274i 
 + m^2*225i + m^3*85i + m^4*15i + m^5*1i + 120i)) - (d^4*(50*m + 35*m^2 + 
10*m^3 + m^4 + 24))/(32*f*(m*274i + m^2*225i + m^3*85i + m^4*15i + m^5*1i 
+ 120i)) + (exp(e*5i + f*x*5i)*(c^4*m*4928i - c^3*d*3840i - c*d^3*2400i + 
d^4*m*264i + c^4*3840i + d^4*768i + c^2*d^2*7680i + c^4*m^2*2272i + c^4*m^ 
3*448i + c^4*m^4*32i + d^4*m^2*324i + d^4*m^3*72i + d^4*m^4*12i + c^2*d^2* 
m*5376i + c*d^3*m^2*816i + c^3*d*m^2*2240i + c*d^3*m^3*336i + c^3*d*m^3*70 
4i + c*d^3*m^4*48i + c^3*d*m^4*64i + c^2*d^2*m^2*3168i + c^2*d^2*m^3*960i 
+ c^2*d^2*m^4*96i + c*d^3*m*1200i + c^3*d*m*832i))/(32*f*(m*274i + m^2*...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1014, normalized size of antiderivative = 5.96 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx =\text {Too large to display} \] Input:

int(cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^4,x)
 

Output:

((sin(e + f*x)*a + a)**m*(sin(e + f*x)**5*d**4*m**4 + 10*sin(e + f*x)**5*d 
**4*m**3 + 35*sin(e + f*x)**5*d**4*m**2 + 50*sin(e + f*x)**5*d**4*m + 24*s 
in(e + f*x)**5*d**4 + 4*sin(e + f*x)**4*c*d**3*m**4 + 44*sin(e + f*x)**4*c 
*d**3*m**3 + 164*sin(e + f*x)**4*c*d**3*m**2 + 244*sin(e + f*x)**4*c*d**3* 
m + 120*sin(e + f*x)**4*c*d**3 + sin(e + f*x)**4*d**4*m**4 + 6*sin(e + f*x 
)**4*d**4*m**3 + 11*sin(e + f*x)**4*d**4*m**2 + 6*sin(e + f*x)**4*d**4*m + 
 6*sin(e + f*x)**3*c**2*d**2*m**4 + 72*sin(e + f*x)**3*c**2*d**2*m**3 + 29 
4*sin(e + f*x)**3*c**2*d**2*m**2 + 468*sin(e + f*x)**3*c**2*d**2*m + 240*s 
in(e + f*x)**3*c**2*d**2 + 4*sin(e + f*x)**3*c*d**3*m**4 + 32*sin(e + f*x) 
**3*c*d**3*m**3 + 68*sin(e + f*x)**3*c*d**3*m**2 + 40*sin(e + f*x)**3*c*d* 
*3*m - 4*sin(e + f*x)**3*d**4*m**3 - 12*sin(e + f*x)**3*d**4*m**2 - 8*sin( 
e + f*x)**3*d**4*m + 4*sin(e + f*x)**2*c**3*d*m**4 + 52*sin(e + f*x)**2*c* 
*3*d*m**3 + 236*sin(e + f*x)**2*c**3*d*m**2 + 428*sin(e + f*x)**2*c**3*d*m 
 + 240*sin(e + f*x)**2*c**3*d + 6*sin(e + f*x)**2*c**2*d**2*m**4 + 60*sin( 
e + f*x)**2*c**2*d**2*m**3 + 174*sin(e + f*x)**2*c**2*d**2*m**2 + 120*sin( 
e + f*x)**2*c**2*d**2*m - 12*sin(e + f*x)**2*c*d**3*m**3 - 72*sin(e + f*x) 
**2*c*d**3*m**2 - 60*sin(e + f*x)**2*c*d**3*m + 12*sin(e + f*x)**2*d**4*m* 
*2 + 12*sin(e + f*x)**2*d**4*m + sin(e + f*x)*c**4*m**4 + 14*sin(e + f*x)* 
c**4*m**3 + 71*sin(e + f*x)*c**4*m**2 + 154*sin(e + f*x)*c**4*m + 120*sin( 
e + f*x)*c**4 + 4*sin(e + f*x)*c**3*d*m**4 + 48*sin(e + f*x)*c**3*d*m**...