\(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\) [697]

Optimal result

Integrand size = 29, antiderivative size = 184 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {8 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac {6 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {6 a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac {8 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac {3 a^3 \sin ^{9+n}(c+d x)}{d (9+n)}-\frac {a^3 \sin ^{10+n}(c+d x)}{d (10+n)} \] Output:

a^3*sin(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)-8*a^3*sin(d*x+ 
c)^(4+n)/d/(4+n)-6*a^3*sin(d*x+c)^(5+n)/d/(5+n)+6*a^3*sin(d*x+c)^(6+n)/d/( 
6+n)+8*a^3*sin(d*x+c)^(7+n)/d/(7+n)-3*a^3*sin(d*x+c)^(9+n)/d/(9+n)-a^3*sin 
(d*x+c)^(10+n)/d/(10+n)
 

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {3 \sin (c+d x)}{2+n}-\frac {8 \sin ^3(c+d x)}{4+n}-\frac {6 \sin ^4(c+d x)}{5+n}+\frac {6 \sin ^5(c+d x)}{6+n}+\frac {8 \sin ^6(c+d x)}{7+n}-\frac {3 \sin ^8(c+d x)}{9+n}-\frac {\sin ^9(c+d x)}{10+n}\right )}{d} \] Input:

Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
 

Output:

(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) - (8*Si 
n[c + d*x]^3)/(4 + n) - (6*Sin[c + d*x]^4)/(5 + n) + (6*Sin[c + d*x]^5)/(6 
 + n) + (8*Sin[c + d*x]^6)/(7 + n) - (3*Sin[c + d*x]^8)/(9 + n) - Sin[c + 
d*x]^9/(10 + n)))/d
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3042, 3315, 99, 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.

\(\displaystyle \int \cos ^7(c+d x) (a \sin (c+d x)+a)^3 \sin ^n(c+d x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^7 (a \sin (c+d x)+a)^3 \sin (c+d x)^ndx\)

\(\Big \downarrow \) 3315

\(\displaystyle \frac {\int \sin ^n(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^6d(a \sin (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (a^9 \sin ^n(c+d x)+3 a^9 \sin ^{n+1}(c+d x)-8 a^9 \sin ^{n+3}(c+d x)-6 a^9 \sin ^{n+4}(c+d x)+6 a^9 \sin ^{n+5}(c+d x)+8 a^9 \sin ^{n+6}(c+d x)-3 a^9 \sin ^{n+8}(c+d x)-a^9 \sin ^{n+9}(c+d x)\right )d(a \sin (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^{10} \sin ^{n+1}(c+d x)}{n+1}+\frac {3 a^{10} \sin ^{n+2}(c+d x)}{n+2}-\frac {8 a^{10} \sin ^{n+4}(c+d x)}{n+4}-\frac {6 a^{10} \sin ^{n+5}(c+d x)}{n+5}+\frac {6 a^{10} \sin ^{n+6}(c+d x)}{n+6}+\frac {8 a^{10} \sin ^{n+7}(c+d x)}{n+7}-\frac {3 a^{10} \sin ^{n+9}(c+d x)}{n+9}-\frac {a^{10} \sin ^{n+10}(c+d x)}{n+10}}{a^7 d}\)

Input:

Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]
 

Output:

((a^10*Sin[c + d*x]^(1 + n))/(1 + n) + (3*a^10*Sin[c + d*x]^(2 + n))/(2 + 
n) - (8*a^10*Sin[c + d*x]^(4 + n))/(4 + n) - (6*a^10*Sin[c + d*x]^(5 + n)) 
/(5 + n) + (6*a^10*Sin[c + d*x]^(6 + n))/(6 + n) + (8*a^10*Sin[c + d*x]^(7 
 + n))/(7 + n) - (3*a^10*Sin[c + d*x]^(9 + n))/(9 + n) - (a^10*Sin[c + d*x 
]^(10 + n))/(10 + n))/(a^7*d)
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

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 3315

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(184)=368\).

Time = 30.30 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.26

method	result	size
parallelrisch	\(\frac {19 \sin \left (d x +c \right )^{n} \left (\frac {\left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (10 d x +10 c \right )}{76}+\frac {25 \left (9+n \right ) \left (1+n \right ) \left (5+n \right ) \left (7+n \right ) \left (n^{3}+\frac {492}{25} n^{2}+\frac {2252}{25} n -\frac {4368}{5}\right ) \cos \left (2 d x +2 c \right )}{38}-\frac {14 \left (2+n \right ) \left (n^{2}+\frac {166}{7} n +\frac {1320}{7}\right ) \left (9+n \right ) \left (1+n \right ) \left (5+n \right ) \left (7+n \right ) \cos \left (4 d x +4 c \right )}{19}-\frac {51 \left (2+n \right ) \left (9+n \right ) \left (1+n \right ) \left (5+n \right ) \left (7+n \right ) \left (4+n \right ) \left (n +\frac {230}{17}\right ) \cos \left (6 d x +6 c \right )}{76}-\frac {5 \left (9+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (8 d x +8 c \right )}{38}+\frac {30 \left (2+n \right ) \left (1+n \right ) \left (10+n \right ) \left (4+n \right ) \left (n^{2}+\frac {108}{5} n +147\right ) \left (6+n \right ) \sin \left (3 d x +3 c \right )}{19}+\frac {10 \left (2+n \right ) \left (1+n \right ) \left (10+n \right ) \left (4+n \right ) \left (n^{2}+\frac {76}{5} n +\frac {63}{5}\right ) \left (6+n \right ) \sin \left (5 d x +5 c \right )}{19}-\frac {5 \left (2+n \right ) \left (1+n \right ) \left (n +\frac {99}{5}\right ) \left (10+n \right ) \left (5+n \right ) \left (4+n \right ) \left (6+n \right ) \sin \left (7 d x +7 c \right )}{38}-\frac {3 \left (10+n \right ) \left (7+n \right ) \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (9 d x +9 c \right )}{38}+\left (2+n \right ) \left (n^{3}+\frac {471}{19} n^{2}+239 n +\frac {28665}{19}\right ) \left (10+n \right ) \left (4+n \right ) \left (6+n \right ) \sin \left (d x +c \right )+\frac {33 \left (9+n \right ) \left (1+n \right ) \left (n^{3}+\frac {260}{11} n^{2}+\frac {2276}{11} n +1072\right ) \left (5+n \right ) \left (7+n \right )}{38}\right ) a^{3}}{128 \left (2+n \right ) \left (1+n \right ) \left (9+n \right ) \left (10+n \right ) \left (4+n \right ) \left (6+n \right ) \left (7+n \right ) \left (5+n \right ) d}\)	\(415\)

Input:

int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

19/128*sin(d*x+c)^n*(1/76*(9+n)*(7+n)*(6+n)*(5+n)*(4+n)*(2+n)*(1+n)*cos(10 
*d*x+10*c)+25/38*(9+n)*(1+n)*(5+n)*(7+n)*(n^3+492/25*n^2+2252/25*n-4368/5) 
*cos(2*d*x+2*c)-14/19*(2+n)*(n^2+166/7*n+1320/7)*(9+n)*(1+n)*(5+n)*(7+n)*c 
os(4*d*x+4*c)-51/76*(2+n)*(9+n)*(1+n)*(5+n)*(7+n)*(4+n)*(n+230/17)*cos(6*d 
*x+6*c)-5/38*(9+n)*(7+n)*(6+n)*(5+n)*(4+n)*(2+n)*(1+n)*cos(8*d*x+8*c)+30/1 
9*(2+n)*(1+n)*(10+n)*(4+n)*(n^2+108/5*n+147)*(6+n)*sin(3*d*x+3*c)+10/19*(2 
+n)*(1+n)*(10+n)*(4+n)*(n^2+76/5*n+63/5)*(6+n)*sin(5*d*x+5*c)-5/38*(2+n)*( 
1+n)*(n+99/5)*(10+n)*(5+n)*(4+n)*(6+n)*sin(7*d*x+7*c)-3/38*(10+n)*(7+n)*(6 
+n)*(5+n)*(4+n)*(2+n)*(1+n)*sin(9*d*x+9*c)+(2+n)*(n^3+471/19*n^2+239*n+286 
65/19)*(10+n)*(4+n)*(6+n)*sin(d*x+c)+33/38*(9+n)*(1+n)*(n^3+260/11*n^2+227 
6/11*n+1072)*(5+n)*(7+n))*a^3/(2+n)/(1+n)/(9+n)/(10+n)/(4+n)/(6+n)/(7+n)/( 
5+n)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (184) = 368\).

Time = 0.16 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.79 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

((a^3*n^7 + 34*a^3*n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 3180 
4*a^3*n^2 + 35844*a^3*n + 15120*a^3)*cos(d*x + c)^10 - 5*(a^3*n^7 + 34*a^3 
*n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 31804*a^3*n^2 + 35844* 
a^3*n + 15120*a^3)*cos(d*x + c)^8 + 192*a^3*n^4 + 4*(a^3*n^7 + 28*a^3*n^6 
+ 304*a^3*n^5 + 1618*a^3*n^4 + 4375*a^3*n^3 + 5554*a^3*n^2 + 2520*a^3*n)*c 
os(d*x + c)^6 + 4224*a^3*n^3 + 31488*a^3*n^2 + 24*(a^3*n^6 + 24*a^3*n^5 + 
208*a^3*n^4 + 786*a^3*n^3 + 1231*a^3*n^2 + 630*a^3*n)*cos(d*x + c)^4 + 879 
36*a^3*n + 60480*a^3 + 96*(a^3*n^5 + 22*a^3*n^4 + 164*a^3*n^3 + 458*a^3*n^ 
2 + 315*a^3*n)*cos(d*x + c)^2 - (3*(a^3*n^7 + 35*a^3*n^6 + 497*a^3*n^5 + 3 
689*a^3*n^4 + 15302*a^3*n^3 + 34916*a^3*n^2 + 39640*a^3*n + 16800*a^3)*cos 
(d*x + c)^8 - 192*a^3*n^4 - 4*(a^3*n^7 + 31*a^3*n^6 + 385*a^3*n^5 + 2485*a 
^3*n^4 + 8974*a^3*n^3 + 18004*a^3*n^2 + 18360*a^3*n + 7200*a^3)*cos(d*x + 
c)^6 - 4224*a^3*n^3 - 31488*a^3*n^2 - 24*(a^3*n^6 + 26*a^3*n^5 + 255*a^3*n 
^4 + 1210*a^3*n^3 + 2924*a^3*n^2 + 3384*a^3*n + 1440*a^3)*cos(d*x + c)^4 - 
 93696*a^3*n - 92160*a^3 - 96*(a^3*n^5 + 23*a^3*n^4 + 186*a^3*n^3 + 652*a^ 
3*n^2 + 968*a^3*n + 480*a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/ 
(d*n^8 + 44*d*n^7 + 812*d*n^6 + 8162*d*n^5 + 48503*d*n^4 + 172634*d*n^3 + 
353884*d*n^2 + 373560*d*n + 151200*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41196 vs. \(2 (158) = 316\).

Time = 84.30 (sec) , antiderivative size = 41196, normalized size of antiderivative = 223.89 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**7*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)
 

Output:

Piecewise((x*(a*sin(c) + a)**3*sin(c)**n*cos(c)**7, Eq(d, 0)), (-a**3*log( 
sin(c + d*x))/d + 48*a**3/(35*d*sin(c + d*x)) - a**3*cos(c + d*x)**2/(2*d* 
sin(c + d*x)**2) + 3*a**3/(8*d*sin(c + d*x)**2) - 24*a**3*cos(c + d*x)**2/ 
(35*d*sin(c + d*x)**3) + 16*a**3/(315*d*sin(c + d*x)**3) + a**3*cos(c + d* 
x)**4/(4*d*sin(c + d*x)**4) - 3*a**3*cos(c + d*x)**2/(8*d*sin(c + d*x)**4) 
 + 18*a**3*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - 8*a**3*cos(c + d*x)**2 
/(105*d*sin(c + d*x)**5) - a**3*cos(c + d*x)**6/(6*d*sin(c + d*x)**6) + 3* 
a**3*cos(c + d*x)**4/(8*d*sin(c + d*x)**6) - 3*a**3*cos(c + d*x)**6/(7*d*s 
in(c + d*x)**7) + 2*a**3*cos(c + d*x)**4/(21*d*sin(c + d*x)**7) - 3*a**3*c 
os(c + d*x)**6/(8*d*sin(c + d*x)**8) - a**3*cos(c + d*x)**6/(9*d*sin(c + d 
*x)**9), Eq(n, -10)), (-3*a**3*log(sin(c + d*x))/d - 16*a**3*sin(c + d*x)/ 
(5*d) - 8*a**3*cos(c + d*x)**2/(5*d*sin(c + d*x)) + 48*a**3/(35*d*sin(c + 
d*x)) - 3*a**3*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + a**3/(8*d*sin(c + d 
*x)**2) + 2*a**3*cos(c + d*x)**4/(5*d*sin(c + d*x)**3) - 24*a**3*cos(c + d 
*x)**2/(35*d*sin(c + d*x)**3) + 3*a**3*cos(c + d*x)**4/(4*d*sin(c + d*x)** 
4) - a**3*cos(c + d*x)**2/(8*d*sin(c + d*x)**4) - a**3*cos(c + d*x)**6/(5* 
d*sin(c + d*x)**5) + 18*a**3*cos(c + d*x)**4/(35*d*sin(c + d*x)**5) - a**3 
*cos(c + d*x)**6/(2*d*sin(c + d*x)**6) + a**3*cos(c + d*x)**4/(8*d*sin(c + 
 d*x)**6) - 3*a**3*cos(c + d*x)**6/(7*d*sin(c + d*x)**7) - a**3*cos(c + d* 
x)**6/(8*d*sin(c + d*x)**8), Eq(n, -9)), (-a**3*tan(c/2 + d*x/2)**18/(8...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 10}}{n + 10} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 9}}{n + 9} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

-(a^3*sin(d*x + c)^(n + 10)/(n + 10) + 3*a^3*sin(d*x + c)^(n + 9)/(n + 9) 
- 8*a^3*sin(d*x + c)^(n + 7)/(n + 7) - 6*a^3*sin(d*x + c)^(n + 6)/(n + 6) 
+ 6*a^3*sin(d*x + c)^(n + 5)/(n + 5) + 8*a^3*sin(d*x + c)^(n + 4)/(n + 4) 
- 3*a^3*sin(d*x + c)^(n + 2)/(n + 2) - a^3*sin(d*x + c)^(n + 1)/(n + 1))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (184) = 368\).

Time = 0.21 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.71 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {\frac {a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7}}{n + 7} + \frac {a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 3 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 10} + \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 9} + \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{7}}{n + 8} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} - \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 3 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 8} - \frac {9 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 7} - \frac {9 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{5}}{n + 6} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac {3 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 3 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 6} + \frac {9 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + 2 \, \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 5} + \frac {9 \, a^{3} e^{\left (n \log \left (\sin \left (d x + c\right )\right ) + \log \left (\sin \left (d x + c\right )\right )\right )} \sin \left (d x + c\right )^{3}}{n + 4} - \frac {a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

-(a^3*sin(d*x + c)^n*sin(d*x + c)^7/(n + 7) + a^3*e^(n*log(sin(d*x + c)) + 
 3*log(sin(d*x + c)))*sin(d*x + c)^7/(n + 10) + 3*a^3*e^(n*log(sin(d*x + c 
)) + 2*log(sin(d*x + c)))*sin(d*x + c)^7/(n + 9) + 3*a^3*e^(n*log(sin(d*x 
+ c)) + log(sin(d*x + c)))*sin(d*x + c)^7/(n + 8) - 3*a^3*sin(d*x + c)^n*s 
in(d*x + c)^5/(n + 5) - 3*a^3*e^(n*log(sin(d*x + c)) + 3*log(sin(d*x + c)) 
)*sin(d*x + c)^5/(n + 8) - 9*a^3*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + 
c)))*sin(d*x + c)^5/(n + 7) - 9*a^3*e^(n*log(sin(d*x + c)) + log(sin(d*x + 
 c)))*sin(d*x + c)^5/(n + 6) + 3*a^3*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) 
 + 3*a^3*e^(n*log(sin(d*x + c)) + 3*log(sin(d*x + c)))*sin(d*x + c)^3/(n + 
 6) + 9*a^3*e^(n*log(sin(d*x + c)) + 2*log(sin(d*x + c)))*sin(d*x + c)^3/( 
n + 5) + 9*a^3*e^(n*log(sin(d*x + c)) + log(sin(d*x + c)))*sin(d*x + c)^3/ 
(n + 4) - a^3*sin(d*x + c)^(n + 4)/(n + 4) - 3*a^3*sin(d*x + c)^(n + 3)/(n 
 + 3) - 3*a^3*sin(d*x + c)^(n + 2)/(n + 2) - a^3*sin(d*x + c)^(n + 1)/(n + 
 1))/d
 

Mupad [B] (verification not implemented)

Time = 42.27 (sec) , antiderivative size = 1130, normalized size of antiderivative = 6.14 \[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^7*sin(c + d*x)^n*(a + a*sin(c + d*x))^3,x)
 

Output:

(3*a^3*sin(c + d*x)^n*(6117676*n + 3058196*n^2 + 755233*n^3 + 109542*n^4 + 
 9800*n^5 + 502*n^6 + 11*n^7 + 3714480))/(256*d*(373560*n + 353884*n^2 + 1 
72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (5* 
a^3*sin(c + d*x)^n*cos(8*c + 8*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 344 
2*n^4 + 472*n^5 + 34*n^6 + n^7 + 15120))/(256*d*(373560*n + 353884*n^2 + 1 
72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^ 
3*sin(c + d*x)^n*cos(10*c + 10*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 344 
2*n^4 + 472*n^5 + 34*n^6 + n^7 + 15120))/(512*d*(373560*n + 353884*n^2 + 1 
72634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (a^ 
3*sin(c + d*x)*sin(c + d*x)^n*(n*16168200i + n^2*7143148i + n^3*1614322i + 
 n^4*215083i + n^5*18019i + n^6*889i + n^7*19i + 13759200i)*1i)/(128*d*(37 
3560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 
 + n^8 + 151200)) - (3*a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(1320260*n + 11 
00668*n^2 + 446515*n^3 + 97426*n^4 + 11608*n^5 + 706*n^6 + 17*n^7 + 579600 
))/(512*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812 
*n^6 + 44*n^7 + n^8 + 151200)) - (a^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(172 
9500*n + 1246276*n^2 + 413653*n^3 + 71710*n^4 + 6760*n^5 + 334*n^6 + 7*n^7 
 + 831600))/(64*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n 
^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^3*sin(c + d*x)^n*cos(2*c + 2*d 
*x)*(122059*n^3 - 2395364*n^2 - 9293340*n + 119842*n^4 + 17176*n^5 + 10...
 

Reduce [F]

\[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{3}d x +3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}d x \right )+3 \left (\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )d x \right )+\int \sin \left (d x +c \right )^{n} \cos \left (d x +c \right )^{7}d x \right ) \] Input:

int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)
 

Output:

a**3*(int(sin(c + d*x)**n*cos(c + d*x)**7*sin(c + d*x)**3,x) + 3*int(sin(c 
 + d*x)**n*cos(c + d*x)**7*sin(c + d*x)**2,x) + 3*int(sin(c + d*x)**n*cos( 
c + d*x)**7*sin(c + d*x),x) + int(sin(c + d*x)**n*cos(c + d*x)**7,x))