\(\int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx\) [910]

Optimal result

Integrand size = 31, antiderivative size = 175 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx=\frac {a^4 (c-d)^4 (c+d \sin (e+f x))^{1+n}}{d^5 f (1+n)}-\frac {4 a^4 (c-d)^3 (c+d \sin (e+f x))^{2+n}}{d^5 f (2+n)}+\frac {6 a^4 (c-d)^2 (c+d \sin (e+f x))^{3+n}}{d^5 f (3+n)}-\frac {4 a^4 (c-d) (c+d \sin (e+f x))^{4+n}}{d^5 f (4+n)}+\frac {a^4 (c+d \sin (e+f x))^{5+n}}{d^5 f (5+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.74 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx=\frac {a^4 (c+d \sin (e+f x))^{1+n} \left (\frac {(c-d)^4}{1+n}-\frac {4 (c-d)^3 (c+d \sin (e+f x))}{2+n}+\frac {6 (c-d)^2 (c+d \sin (e+f x))^2}{3+n}-\frac {4 (c-d) (c+d \sin (e+f x))^3}{4+n}+\frac {(c+d \sin (e+f x))^4}{5+n}\right )}{d^5 f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 3312, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3312

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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. \(385\) vs. \(2(175)=350\).

Time = 45.05 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.21

method	result	size
parallelrisch	\(-\frac {24 \left (c +d \sin \left (f x +e \right )\right )^{n} \left (\frac {\left (\frac {2 \left (5+n \right ) \left (3+n \right )^{2} d^{3}}{3}+\frac {7 n c \left (n^{2}+\frac {59}{7} n +18\right ) d^{2}}{12}-c^{2} n \left (5+n \right ) d +c^{3} n \right ) \left (1+n \right ) d^{2} \cos \left (2 f x +2 e \right )}{4}-\frac {\left (1+n \right ) \left (2+n \right ) \left (\left (-\frac {29}{16} n^{2}-\frac {251}{16} n -\frac {135}{4}\right ) d^{2}-c n \left (5+n \right ) d +c^{2} n \right ) d^{3} \sin \left (3 f x +3 e \right )}{24}-\frac {\left (1+n \right ) \left (\left (4 n +20\right ) d +c n \right ) \left (3+n \right ) \left (2+n \right ) d^{4} \cos \left (4 f x +4 e \right )}{192}-\frac {d^{5} \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 f x +5 e \right )}{384}+\left (-\frac {49 \left (n^{2}+\frac {300}{49} n +\frac {45}{7}\right ) \left (2+n \right ) \left (4+n \right ) d^{4}}{192}-\frac {7 \left (n^{2}+\frac {37}{7} n +\frac {54}{7}\right ) n c \left (5+n \right ) d^{3}}{24}+\frac {5 n \,c^{2} \left (n^{2}+\frac {39}{5} n +\frac {82}{5}\right ) d^{2}}{8}-c^{3} n \left (5+n \right ) d +c^{4} n \right ) d \sin \left (f x +e \right )-\frac {7 \left (1+n \right ) \left (3+n \right ) \left (n +\frac {22}{7}\right ) \left (5+n \right ) d^{5}}{48}-\frac {35 c \left (n^{4}+\frac {74}{7} n^{3}+\frac {1297}{35} n^{2}+\frac {346}{7} n +\frac {192}{7}\right ) d^{4}}{192}+\frac {5 c^{2} \left (5+n \right ) \left (n^{2}+\frac {17}{5} n +\frac {24}{5}\right ) d^{3}}{12}-\frac {3 \left (n^{2}+\frac {19}{3} n +\frac {40}{3}\right ) c^{3} d^{2}}{4}+c^{4} \left (5+n \right ) d -c^{5}\right ) a^{4}}{d^{5} f \left (1+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right )}\)	\(386\)
derivativedivides	\(\frac {a^{4} \sin \left (f x +e \right )^{5} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f \left (5+n \right )}+\frac {a^{4} c \left (d^{4} n^{4}-4 c \,d^{3} n^{3}+14 d^{4} n^{3}+12 c^{2} d^{2} n^{2}-48 c \,d^{3} n^{2}+71 d^{4} n^{2}-24 c^{3} d n +108 c^{2} d^{2} n -188 c \,d^{3} n +154 d^{4} n +24 c^{4}-120 c^{3} d +240 c^{2} d^{2}-240 c \,d^{3}+120 d^{4}\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{5} f \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {a^{4} \left (c n +4 n d +20 d \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d f \left (n^{2}+9 n +20\right )}-\frac {a^{4} \left (-4 c \,d^{3} n^{4}-d^{4} n^{4}+12 c^{2} d^{2} n^{3}-48 c \,d^{3} n^{3}-14 d^{4} n^{3}-24 c^{3} d \,n^{2}+108 c^{2} d^{2} n^{2}-188 c \,d^{3} n^{2}-71 d^{4} n^{2}+24 c^{4} n -120 c^{3} d n +240 c^{2} d^{2} n -240 c \,d^{3} n -154 d^{4} n -120 d^{4}\right ) \sin \left (f x +e \right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) f}-\frac {2 \left (-2 c d \,n^{2}-3 d^{2} n^{2}+2 c^{2} n -10 c d n -27 d^{2} n -60 d^{2}\right ) a^{4} \sin \left (f x +e \right )^{3} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{2} f \left (n^{3}+12 n^{2}+47 n +60\right )}+\frac {2 \left (3 c \,d^{2} n^{3}+2 d^{3} n^{3}-6 c^{2} d \,n^{2}+27 c \,d^{2} n^{2}+24 d^{3} n^{2}+6 c^{3} n -30 c^{2} d n +60 c \,d^{2} n +94 d^{3} n +120 d^{3}\right ) a^{4} \sin \left (f x +e \right )^{2} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{3} f \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}\)	\(644\)
default	\(\frac {a^{4} \sin \left (f x +e \right )^{5} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f \left (5+n \right )}+\frac {a^{4} c \left (d^{4} n^{4}-4 c \,d^{3} n^{3}+14 d^{4} n^{3}+12 c^{2} d^{2} n^{2}-48 c \,d^{3} n^{2}+71 d^{4} n^{2}-24 c^{3} d n +108 c^{2} d^{2} n -188 c \,d^{3} n +154 d^{4} n +24 c^{4}-120 c^{3} d +240 c^{2} d^{2}-240 c \,d^{3}+120 d^{4}\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{5} f \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {a^{4} \left (c n +4 n d +20 d \right ) \sin \left (f x +e \right )^{4} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d f \left (n^{2}+9 n +20\right )}-\frac {a^{4} \left (-4 c \,d^{3} n^{4}-d^{4} n^{4}+12 c^{2} d^{2} n^{3}-48 c \,d^{3} n^{3}-14 d^{4} n^{3}-24 c^{3} d \,n^{2}+108 c^{2} d^{2} n^{2}-188 c \,d^{3} n^{2}-71 d^{4} n^{2}+24 c^{4} n -120 c^{3} d n +240 c^{2} d^{2} n -240 c \,d^{3} n -154 d^{4} n -120 d^{4}\right ) \sin \left (f x +e \right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) f}-\frac {2 \left (-2 c d \,n^{2}-3 d^{2} n^{2}+2 c^{2} n -10 c d n -27 d^{2} n -60 d^{2}\right ) a^{4} \sin \left (f x +e \right )^{3} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{2} f \left (n^{3}+12 n^{2}+47 n +60\right )}+\frac {2 \left (3 c \,d^{2} n^{3}+2 d^{3} n^{3}-6 c^{2} d \,n^{2}+27 c \,d^{2} n^{2}+24 d^{3} n^{2}+6 c^{3} n -30 c^{2} d n +60 c \,d^{2} n +94 d^{3} n +120 d^{3}\right ) a^{4} \sin \left (f x +e \right )^{2} {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{3} f \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}\)	\(644\)

Input:

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

Output:

-24*(c+d*sin(f*x+e))^n*(1/4*(2/3*(5+n)*(3+n)^2*d^3+7/12*n*c*(n^2+59/7*n+18 
)*d^2-c^2*n*(5+n)*d+c^3*n)*(1+n)*d^2*cos(2*f*x+2*e)-1/24*(1+n)*(2+n)*((-29 
/16*n^2-251/16*n-135/4)*d^2-c*n*(5+n)*d+c^2*n)*d^3*sin(3*f*x+3*e)-1/192*(1 
+n)*((4*n+20)*d+c*n)*(3+n)*(2+n)*d^4*cos(4*f*x+4*e)-1/384*d^5*(4+n)*(3+n)* 
(2+n)*(1+n)*sin(5*f*x+5*e)+(-49/192*(n^2+300/49*n+45/7)*(2+n)*(4+n)*d^4-7/ 
24*(n^2+37/7*n+54/7)*n*c*(5+n)*d^3+5/8*n*c^2*(n^2+39/5*n+82/5)*d^2-c^3*n*( 
5+n)*d+c^4*n)*d*sin(f*x+e)-7/48*(1+n)*(3+n)*(n+22/7)*(5+n)*d^5-35/192*c*(n 
^4+74/7*n^3+1297/35*n^2+346/7*n+192/7)*d^4+5/12*c^2*(5+n)*(n^2+17/5*n+24/5 
)*d^3-3/4*(n^2+19/3*n+40/3)*c^3*d^2+c^4*(5+n)*d-c^5)*a^4/d^5/f/(1+n)/(5+n) 
/(4+n)/(3+n)/(2+n)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (175) = 350\).

Time = 0.16 (sec) , antiderivative size = 902, normalized size of antiderivative = 5.15 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx =\text {Too large to display} \] Input:

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

Output:

(24*a^4*c^5 - 120*a^4*c^4*d + 240*a^4*c^3*d^2 - 240*a^4*c^2*d^3 + 120*a^4* 
c*d^4 + 360*a^4*d^5 + 8*(a^4*c*d^4 + a^4*d^5)*n^4 + (120*a^4*d^5 + (a^4*c* 
d^4 + 4*a^4*d^5)*n^4 + 2*(3*a^4*c*d^4 + 22*a^4*d^5)*n^3 + (11*a^4*c*d^4 + 
164*a^4*d^5)*n^2 + 2*(3*a^4*c*d^4 + 122*a^4*d^5)*n)*cos(f*x + e)^4 - 16*(a 
^4*c^2*d^3 - 5*a^4*c*d^4 - 6*a^4*d^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2* 
d^3 + 32*a^4*c*d^4 + 50*a^4*d^5)*n^2 - 4*(120*a^4*d^5 + (2*a^4*c*d^4 + 3*a 
^4*d^5)*n^4 - (3*a^4*c^2*d^3 - 18*a^4*c*d^4 - 35*a^4*d^5)*n^3 + (3*a^4*c^3 
*d^2 - 18*a^4*c^2*d^3 + 49*a^4*c*d^4 + 141*a^4*d^5)*n^2 + (3*a^4*c^3*d^2 - 
 15*a^4*c^2*d^3 + 33*a^4*c*d^4 + 229*a^4*d^5)*n)*cos(f*x + e)^2 - 8*(3*a^4 
*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^4*c*d^4 - 84*a^4*d^5)*n + 
(384*a^4*d^5 + 8*(a^4*c*d^4 + a^4*d^5)*n^4 + (a^4*d^5*n^4 + 10*a^4*d^5*n^3 
 + 35*a^4*d^5*n^2 + 50*a^4*d^5*n + 24*a^4*d^5)*cos(f*x + e)^4 - 16*(a^4*c^ 
2*d^3 - 5*a^4*c*d^4 - 6*a^4*d^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 
 32*a^4*c*d^4 + 50*a^4*d^5)*n^2 - 4*(72*a^4*d^5 + (a^4*c*d^4 + 2*a^4*d^5)* 
n^4 - (a^4*c^2*d^3 - 8*a^4*c*d^4 - 23*a^4*d^5)*n^3 - (3*a^4*c^2*d^3 - 17*a 
^4*c*d^4 - 91*a^4*d^5)*n^2 - 2*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 71*a^4*d^5)*n) 
*cos(f*x + e)^2 - 8*(3*a^4*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^ 
4*c*d^4 - 84*a^4*d^5)*n)*sin(f*x + e))*(d*sin(f*x + e) + c)^n/(d^5*f*n^5 + 
 15*d^5*f*n^4 + 85*d^5*f*n^3 + 225*d^5*f*n^2 + 274*d^5*f*n + 120*d^5*f)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11900 vs. \(2 (150) = 300\).

Time = 21.89 (sec) , antiderivative size = 11900, normalized size of antiderivative = 68.00 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((c**n*(a**4*sin(e + f*x)**5/(5*f) + a**4*sin(e + f*x)**4/f + 2*a 
**4*sin(e + f*x)**3/f + 2*a**4*sin(e + f*x)**2/f + a**4*sin(e + f*x)/f), E 
q(d, 0)), (x*(c + d*sin(e))**n*(a*sin(e) + a)**4*cos(e), Eq(f, 0)), (12*a* 
*4*c**4*log(c/d + sin(e + f*x))/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f 
*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d* 
*9*f*sin(e + f*x)**4) + 25*a**4*c**4/(12*c**4*d**5*f + 48*c**3*d**6*f*sin( 
e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 
12*d**9*f*sin(e + f*x)**4) + 48*a**4*c**3*d*log(c/d + sin(e + f*x))*sin(e 
+ f*x)/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin( 
e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 8 
8*a**4*c**3*d*sin(e + f*x)/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 
 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f* 
sin(e + f*x)**4) - 12*a**4*c**3*d/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + 
 f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12* 
d**9*f*sin(e + f*x)**4) + 72*a**4*c**2*d**2*log(c/d + sin(e + f*x))*sin(e 
+ f*x)**2/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*s 
in(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) 
+ 108*a**4*c**2*d**2*sin(e + f*x)**2/(12*c**4*d**5*f + 48*c**3*d**6*f*sin( 
e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 
12*d**9*f*sin(e + f*x)**4) - 48*a**4*c**2*d**2*sin(e + f*x)/(12*c**4*d*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (175) = 350\).

Time = 0.05 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.78 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx=\frac {\frac {4 \, {\left (d^{2} {\left (n + 1\right )} \sin \left (f x + e\right )^{2} + c d n \sin \left (f x + e\right ) - c^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{4}}{d {\left (n + 1\right )}} + \frac {6 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} \sin \left (f x + e\right )^{3} + {\left (n^{2} + n\right )} c d^{2} \sin \left (f x + e\right )^{2} - 2 \, c^{2} d n \sin \left (f x + e\right ) + 2 \, c^{3}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {4 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} \sin \left (f x + e\right )^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} \sin \left (f x + e\right )^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} \sin \left (f x + e\right )^{2} + 6 \, c^{3} d n \sin \left (f x + e\right ) - 6 \, c^{4}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{5} \sin \left (f x + e\right )^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c d^{4} \sin \left (f x + e\right )^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{2} d^{3} \sin \left (f x + e\right )^{3} + 12 \, {\left (n^{2} + n\right )} c^{3} d^{2} \sin \left (f x + e\right )^{2} - 24 \, c^{4} d n \sin \left (f x + e\right ) + 24 \, c^{5}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{5}}}{f} \] Input:

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

Output:

(4*(d^2*(n + 1)*sin(f*x + e)^2 + c*d*n*sin(f*x + e) - c^2)*(d*sin(f*x + e) 
 + c)^n*a^4/((n^2 + 3*n + 2)*d^2) + (d*sin(f*x + e) + c)^(n + 1)*a^4/(d*(n 
 + 1)) + 6*((n^2 + 3*n + 2)*d^3*sin(f*x + e)^3 + (n^2 + n)*c*d^2*sin(f*x + 
 e)^2 - 2*c^2*d*n*sin(f*x + e) + 2*c^3)*(d*sin(f*x + e) + c)^n*a^4/((n^3 + 
 6*n^2 + 11*n + 6)*d^3) + 4*((n^3 + 6*n^2 + 11*n + 6)*d^4*sin(f*x + e)^4 + 
 (n^3 + 3*n^2 + 2*n)*c*d^3*sin(f*x + e)^3 - 3*(n^2 + n)*c^2*d^2*sin(f*x + 
e)^2 + 6*c^3*d*n*sin(f*x + e) - 6*c^4)*(d*sin(f*x + e) + c)^n*a^4/((n^4 + 
10*n^3 + 35*n^2 + 50*n + 24)*d^4) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d 
^5*sin(f*x + e)^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*sin(f*x + e)^4 - 4* 
(n^3 + 3*n^2 + 2*n)*c^2*d^3*sin(f*x + e)^3 + 12*(n^2 + n)*c^3*d^2*sin(f*x 
+ e)^2 - 24*c^4*d*n*sin(f*x + e) + 24*c^5)*(d*sin(f*x + e) + c)^n*a^4/((n^ 
5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5))/f
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (175) = 350\).

Time = 0.14 (sec) , antiderivative size = 1180, normalized size of antiderivative = 6.74 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx=\text {Too large to display} \] Input:

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

Output:

(((d*sin(f*x + e) + c)^n*d^5*n^4*sin(f*x + e)^5 + (d*sin(f*x + e) + c)^n*c 
*d^4*n^4*sin(f*x + e)^4 + 10*(d*sin(f*x + e) + c)^n*d^5*n^3*sin(f*x + e)^5 
 + 6*(d*sin(f*x + e) + c)^n*c*d^4*n^3*sin(f*x + e)^4 + 35*(d*sin(f*x + e) 
+ c)^n*d^5*n^2*sin(f*x + e)^5 - 4*(d*sin(f*x + e) + c)^n*c^2*d^3*n^3*sin(f 
*x + e)^3 + 11*(d*sin(f*x + e) + c)^n*c*d^4*n^2*sin(f*x + e)^4 + 50*(d*sin 
(f*x + e) + c)^n*d^5*n*sin(f*x + e)^5 - 12*(d*sin(f*x + e) + c)^n*c^2*d^3* 
n^2*sin(f*x + e)^3 + 6*(d*sin(f*x + e) + c)^n*c*d^4*n*sin(f*x + e)^4 + 24* 
(d*sin(f*x + e) + c)^n*d^5*sin(f*x + e)^5 + 12*(d*sin(f*x + e) + c)^n*c^3* 
d^2*n^2*sin(f*x + e)^2 - 8*(d*sin(f*x + e) + c)^n*c^2*d^3*n*sin(f*x + e)^3 
 + 12*(d*sin(f*x + e) + c)^n*c^3*d^2*n*sin(f*x + e)^2 - 24*(d*sin(f*x + e) 
 + c)^n*c^4*d*n*sin(f*x + e) + 24*(d*sin(f*x + e) + c)^n*c^5)*a^4/(d^5*n^5 
 + 15*d^5*n^4 + 85*d^5*n^3 + 225*d^5*n^2 + 274*d^5*n + 120*d^5) + 4*((d*si 
n(f*x + e) + c)^n*d^4*n^3*sin(f*x + e)^4 + (d*sin(f*x + e) + c)^n*c*d^3*n^ 
3*sin(f*x + e)^3 + 6*(d*sin(f*x + e) + c)^n*d^4*n^2*sin(f*x + e)^4 + 3*(d* 
sin(f*x + e) + c)^n*c*d^3*n^2*sin(f*x + e)^3 + 11*(d*sin(f*x + e) + c)^n*d 
^4*n*sin(f*x + e)^4 - 3*(d*sin(f*x + e) + c)^n*c^2*d^2*n^2*sin(f*x + e)^2 
+ 2*(d*sin(f*x + e) + c)^n*c*d^3*n*sin(f*x + e)^3 + 6*(d*sin(f*x + e) + c) 
^n*d^4*sin(f*x + e)^4 - 3*(d*sin(f*x + e) + c)^n*c^2*d^2*n*sin(f*x + e)^2 
+ 6*(d*sin(f*x + e) + c)^n*c^3*d*n*sin(f*x + e) - 6*(d*sin(f*x + e) + c)^n 
*c^4)*a^4/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4) + 6*(...
 

Mupad [B] (verification not implemented)

Time = 43.05 (sec) , antiderivative size = 863, normalized size of antiderivative = 4.93 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx =\text {Too large to display} \] Input:

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

Output:

(a^4*sin(5*e + 5*f*x)*(c + d*sin(e + f*x))^n*(50*n + 35*n^2 + 10*n^3 + n^4 
 + 24)*1i)/(16*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 120i)) 
+ (a^4*(c + d*sin(e + f*x))^n*(c*d^4*960i - c^4*d*960i + d^5*n*2444i + c^5 
*192i + d^5*1320i - c^2*d^3*1920i + c^3*d^2*1920i + d^5*n^2*1436i + d^5*n^ 
3*340i + d^5*n^4*28i - c^2*d^3*n*1744i + c^3*d^2*n*912i + c*d^4*n^2*1297i 
+ c*d^4*n^3*370i + c*d^4*n^4*35i - c^2*d^3*n^2*672i + c^3*d^2*n^2*144i - c 
^2*d^3*n^3*80i + c*d^4*n*1730i - c^4*d*n*192i))/(8*d^5*f*(n*274i + n^2*225 
i + n^3*85i + n^4*15i + n^5*1i + 120i)) + (a^4*sin(e + f*x)*(c + d*sin(e + 
 f*x))^n*(4290*d^4*n - 192*c^4*n + 2520*d^4 + 2507*d^4*n^2 + 594*d^4*n^3 + 
 49*d^4*n^4 - 1968*c^2*d^2*n + 1912*c*d^3*n^2 + 192*c^3*d*n^2 + 576*c*d^3* 
n^3 + 56*c*d^3*n^4 - 936*c^2*d^2*n^2 - 120*c^2*d^2*n^3 + 2160*c*d^3*n + 96 
0*c^3*d*n)*1i)/(8*d^4*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 
120i)) + (a^4*cos(4*e + 4*f*x)*(c + d*sin(e + f*x))^n*(d*20i + c*n*1i + d* 
n*4i)*(11*n + 6*n^2 + n^3 + 6))/(8*d*f*(n*274i + n^2*225i + n^3*85i + n^4* 
15i + n^5*1i + 120i)) - (a^4*sin(3*e + 3*f*x)*(c + d*sin(e + f*x))^n*(3*n 
+ n^2 + 2)*(251*d^2*n - 16*c^2*n + 540*d^2 + 29*d^2*n^2 + 80*c*d*n + 16*c* 
d*n^2)*1i)/(16*d^2*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 120 
i)) - (a^4*cos(2*e + 2*f*x)*(n + 1)*(c + d*sin(e + f*x))^n*(c^3*n*12i + d^ 
3*n*312i + d^3*360i + d^3*n^2*88i + d^3*n^3*8i + c*d^2*n^2*59i - c^2*d*n^2 
*12i + c*d^2*n^3*7i + c*d^2*n*126i - c^2*d*n*60i))/(2*d^3*f*(n*274i + n...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1030, normalized size of antiderivative = 5.89 \[ \int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx =\text {Too large to display} \] Input:

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

Output:

((sin(e + f*x)*d + c)**n*a**4*(sin(e + f*x)**5*d**5*n**4 + 10*sin(e + f*x) 
**5*d**5*n**3 + 35*sin(e + f*x)**5*d**5*n**2 + 50*sin(e + f*x)**5*d**5*n + 
 24*sin(e + f*x)**5*d**5 + sin(e + f*x)**4*c*d**4*n**4 + 6*sin(e + f*x)**4 
*c*d**4*n**3 + 11*sin(e + f*x)**4*c*d**4*n**2 + 6*sin(e + f*x)**4*c*d**4*n 
 + 4*sin(e + f*x)**4*d**5*n**4 + 44*sin(e + f*x)**4*d**5*n**3 + 164*sin(e 
+ f*x)**4*d**5*n**2 + 244*sin(e + f*x)**4*d**5*n + 120*sin(e + f*x)**4*d** 
5 - 4*sin(e + f*x)**3*c**2*d**3*n**3 - 12*sin(e + f*x)**3*c**2*d**3*n**2 - 
 8*sin(e + f*x)**3*c**2*d**3*n + 4*sin(e + f*x)**3*c*d**4*n**4 + 32*sin(e 
+ f*x)**3*c*d**4*n**3 + 68*sin(e + f*x)**3*c*d**4*n**2 + 40*sin(e + f*x)** 
3*c*d**4*n + 6*sin(e + f*x)**3*d**5*n**4 + 72*sin(e + f*x)**3*d**5*n**3 + 
294*sin(e + f*x)**3*d**5*n**2 + 468*sin(e + f*x)**3*d**5*n + 240*sin(e + f 
*x)**3*d**5 + 12*sin(e + f*x)**2*c**3*d**2*n**2 + 12*sin(e + f*x)**2*c**3* 
d**2*n - 12*sin(e + f*x)**2*c**2*d**3*n**3 - 72*sin(e + f*x)**2*c**2*d**3* 
n**2 - 60*sin(e + f*x)**2*c**2*d**3*n + 6*sin(e + f*x)**2*c*d**4*n**4 + 60 
*sin(e + f*x)**2*c*d**4*n**3 + 174*sin(e + f*x)**2*c*d**4*n**2 + 120*sin(e 
 + f*x)**2*c*d**4*n + 4*sin(e + f*x)**2*d**5*n**4 + 52*sin(e + f*x)**2*d** 
5*n**3 + 236*sin(e + f*x)**2*d**5*n**2 + 428*sin(e + f*x)**2*d**5*n + 240* 
sin(e + f*x)**2*d**5 - 24*sin(e + f*x)*c**4*d*n + 24*sin(e + f*x)*c**3*d** 
2*n**2 + 120*sin(e + f*x)*c**3*d**2*n - 12*sin(e + f*x)*c**2*d**3*n**3 - 1 
08*sin(e + f*x)*c**2*d**3*n**2 - 240*sin(e + f*x)*c**2*d**3*n + 4*sin(e...