\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [644]

Optimal result

Integrand size = 29, antiderivative size = 129 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {23 x}{16 a^3}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {7 \cos ^3(c+d x)}{3 a^3 d}-\frac {3 \cos ^5(c+d x)}{5 a^3 d}+\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {23 \cos (c+d x) \sin ^3(c+d x)}{24 a^3 d}+\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^3 d} \] Output:

-23/16*x/a^3-4*cos(d*x+c)/a^3/d+7/3*cos(d*x+c)^3/a^3/d-3/5*cos(d*x+c)^5/a^ 
3/d+23/16*cos(d*x+c)*sin(d*x+c)/a^3/d+23/24*cos(d*x+c)*sin(d*x+c)^3/a^3/d+ 
1/6*cos(d*x+c)*sin(d*x+c)^5/a^3/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(129)=258\).

Time = 2.79 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-3 (3+920 d x) \cos \left (\frac {c}{2}\right )-2520 \cos \left (\frac {c}{2}+d x\right )-2520 \cos \left (\frac {3 c}{2}+d x\right )+945 \cos \left (\frac {3 c}{2}+2 d x\right )-945 \cos \left (\frac {5 c}{2}+2 d x\right )+380 \cos \left (\frac {5 c}{2}+3 d x\right )+380 \cos \left (\frac {7 c}{2}+3 d x\right )-135 \cos \left (\frac {7 c}{2}+4 d x\right )+135 \cos \left (\frac {9 c}{2}+4 d x\right )-36 \cos \left (\frac {9 c}{2}+5 d x\right )-36 \cos \left (\frac {11 c}{2}+5 d x\right )+5 \cos \left (\frac {11 c}{2}+6 d x\right )-5 \cos \left (\frac {13 c}{2}+6 d x\right )+9 \sin \left (\frac {c}{2}\right )-2760 d x \sin \left (\frac {c}{2}\right )+2520 \sin \left (\frac {c}{2}+d x\right )-2520 \sin \left (\frac {3 c}{2}+d x\right )+945 \sin \left (\frac {3 c}{2}+2 d x\right )+945 \sin \left (\frac {5 c}{2}+2 d x\right )-380 \sin \left (\frac {5 c}{2}+3 d x\right )+380 \sin \left (\frac {7 c}{2}+3 d x\right )-135 \sin \left (\frac {7 c}{2}+4 d x\right )-135 \sin \left (\frac {9 c}{2}+4 d x\right )+36 \sin \left (\frac {9 c}{2}+5 d x\right )-36 \sin \left (\frac {11 c}{2}+5 d x\right )+5 \sin \left (\frac {11 c}{2}+6 d x\right )+5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
 

Output:

(-3*(3 + 920*d*x)*Cos[c/2] - 2520*Cos[c/2 + d*x] - 2520*Cos[(3*c)/2 + d*x] 
 + 945*Cos[(3*c)/2 + 2*d*x] - 945*Cos[(5*c)/2 + 2*d*x] + 380*Cos[(5*c)/2 + 
 3*d*x] + 380*Cos[(7*c)/2 + 3*d*x] - 135*Cos[(7*c)/2 + 4*d*x] + 135*Cos[(9 
*c)/2 + 4*d*x] - 36*Cos[(9*c)/2 + 5*d*x] - 36*Cos[(11*c)/2 + 5*d*x] + 5*Co 
s[(11*c)/2 + 6*d*x] - 5*Cos[(13*c)/2 + 6*d*x] + 9*Sin[c/2] - 2760*d*x*Sin[ 
c/2] + 2520*Sin[c/2 + d*x] - 2520*Sin[(3*c)/2 + d*x] + 945*Sin[(3*c)/2 + 2 
*d*x] + 945*Sin[(5*c)/2 + 2*d*x] - 380*Sin[(5*c)/2 + 3*d*x] + 380*Sin[(7*c 
)/2 + 3*d*x] - 135*Sin[(7*c)/2 + 4*d*x] - 135*Sin[(9*c)/2 + 4*d*x] + 36*Si 
n[(9*c)/2 + 5*d*x] - 36*Sin[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d*x] + 
5*Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 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.

\(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^6(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^6}{(a \sin (c+d x)+a)^3}dx\)

\(\Big \downarrow \) 3348

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3236

\(\displaystyle \frac {\int \left (-a^3 \sin ^6(c+d x)+3 a^3 \sin ^5(c+d x)-3 a^3 \sin ^4(c+d x)+a^3 \sin ^3(c+d x)\right )dx}{a^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {7 a^3 \cos ^3(c+d x)}{3 d}-\frac {4 a^3 \cos (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {23 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}+\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {23 a^3 x}{16}}{a^6}\)

Input:

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]
 

Output:

((-23*a^3*x)/16 - (4*a^3*Cos[c + d*x])/d + (7*a^3*Cos[c + d*x]^3)/(3*d) - 
(3*a^3*Cos[c + d*x]^5)/(5*d) + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + 
 (23*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d 
*x]^5)/(6*d))/a^6
 

Defintions of rubi rules used

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 3236

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + 
f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt 
Q[m, 0] && RationalQ[n]
 

rule 3348

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) 
 + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m)   Int[(d* 
Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, 
 x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60

method	result	size
parallelrisch	\(\frac {-1380 d x +5 \sin \left (6 d x +6 c \right )-135 \sin \left (4 d x +4 c \right )+945 \sin \left (2 d x +2 c \right )-36 \cos \left (5 d x +5 c \right )+380 \cos \left (3 d x +3 c \right )-2520 \cos \left (d x +c \right )-2176}{960 d \,a^{3}}\)	\(78\)
risch	\(-\frac {23 x}{16 a^{3}}-\frac {21 \cos \left (d x +c \right )}{8 a^{3} d}+\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{3}}-\frac {3 \cos \left (5 d x +5 c \right )}{80 a^{3} d}-\frac {9 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}+\frac {19 \cos \left (3 d x +3 c \right )}{48 a^{3} d}+\frac {63 \sin \left (2 d x +2 c \right )}{64 d \,a^{3}}\)	\(107\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{128}-\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{384}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}-\frac {75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}+\frac {75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{384}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}-\frac {17}{60}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\)	\(168\)
default	\(\frac {\frac {16 \left (-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{128}-\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{384}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4}-\frac {75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}+\frac {75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{384}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{10}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}-\frac {17}{60}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\)	\(168\)

Input:

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/960*(-1380*d*x+5*sin(6*d*x+6*c)-135*sin(4*d*x+4*c)+945*sin(2*d*x+2*c)-36 
*cos(5*d*x+5*c)+380*cos(3*d*x+3*c)-2520*cos(d*x+c)-2176)/d/a^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {144 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} + 345 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 62 \, \cos \left (d x + c\right )^{3} + 123 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, a^{3} d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

-1/240*(144*cos(d*x + c)^5 - 560*cos(d*x + c)^3 + 345*d*x - 5*(8*cos(d*x + 
 c)^5 - 62*cos(d*x + c)^3 + 123*cos(d*x + c))*sin(d*x + c) + 960*cos(d*x + 
 c))/(a^3*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2404 vs. \(2 (122) = 244\).

Time = 136.74 (sec) , antiderivative size = 2404, normalized size of antiderivative = 18.64 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+a*sin(d*x+c))**3,x)
 

Output:

Piecewise((-345*d*x*tan(c/2 + d*x/2)**12/(240*a**3*d*tan(c/2 + d*x/2)**12 
+ 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 480 
0*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3 
*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 2070*d*x*tan(c/2 + d*x/2)**10/(240* 
a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3 
*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan 
(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 5175*d* 
x*tan(c/2 + d*x/2)**8/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c 
/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + 
d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2) 
**2 + 240*a**3*d) - 6900*d*x*tan(c/2 + d*x/2)**6/(240*a**3*d*tan(c/2 + d*x 
/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)* 
*8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1 
440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 5175*d*x*tan(c/2 + d*x/2)** 
4/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 36 
00*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a** 
3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 
2070*d*x*tan(c/2 + d*x/2)**2/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3* 
d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan 
(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (117) = 234\).

Time = 0.12 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.89 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3264 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {7680 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2250 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5440 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2250 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {1955 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {345 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 544}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {345 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

1/120*((345*sin(d*x + c)/(cos(d*x + c) + 1) - 3264*sin(d*x + c)^2/(cos(d*x 
 + c) + 1)^2 + 1955*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 7680*sin(d*x + c 
)^4/(cos(d*x + c) + 1)^4 + 2250*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5440 
*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 2250*sin(d*x + c)^7/(cos(d*x + c) + 
 1)^7 - 480*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 1955*sin(d*x + c)^9/(cos 
(d*x + c) + 1)^9 - 345*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 544)/(a^3 + 
 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^4/(cos(d* 
x + c) + 1)^4 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d* 
x + c)^8/(cos(d*x + c) + 1)^8 + 6*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^1 
0 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 345*arctan(sin(d*x + c)/( 
cos(d*x + c) + 1))/a^3)/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {345 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3264 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 544\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

-1/240*(345*(d*x + c)/a^3 + 2*(345*tan(1/2*d*x + 1/2*c)^11 + 1955*tan(1/2* 
d*x + 1/2*c)^9 + 480*tan(1/2*d*x + 1/2*c)^8 + 2250*tan(1/2*d*x + 1/2*c)^7 
+ 5440*tan(1/2*d*x + 1/2*c)^6 - 2250*tan(1/2*d*x + 1/2*c)^5 + 7680*tan(1/2 
*d*x + 1/2*c)^4 - 1955*tan(1/2*d*x + 1/2*c)^3 + 3264*tan(1/2*d*x + 1/2*c)^ 
2 - 345*tan(1/2*d*x + 1/2*c) + 544)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^3))/ 
d
 

Mupad [B] (verification not implemented)

Time = 36.99 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {23\,x}{16\,a^3}-\frac {\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {75\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {136\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {23\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {68}{15}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:

int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + a*sin(c + d*x))^3,x)
 

Output:

- (23*x)/(16*a^3) - ((136*tan(c/2 + (d*x)/2)^2)/5 - (23*tan(c/2 + (d*x)/2) 
)/8 - (391*tan(c/2 + (d*x)/2)^3)/24 + 64*tan(c/2 + (d*x)/2)^4 - (75*tan(c/ 
2 + (d*x)/2)^5)/4 + (136*tan(c/2 + (d*x)/2)^6)/3 + (75*tan(c/2 + (d*x)/2)^ 
7)/4 + 4*tan(c/2 + (d*x)/2)^8 + (391*tan(c/2 + (d*x)/2)^9)/24 + (23*tan(c/ 
2 + (d*x)/2)^11)/8 + 68/15)/(a^3*d*(tan(c/2 + (d*x)/2)^2 + 1)^6)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+230 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-272 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+345 \cos \left (d x +c \right ) \sin \left (d x +c \right )-544 \cos \left (d x +c \right )-345 d x +544}{240 a^{3} d} \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x)
 

Output:

(40*cos(c + d*x)*sin(c + d*x)**5 - 144*cos(c + d*x)*sin(c + d*x)**4 + 230* 
cos(c + d*x)*sin(c + d*x)**3 - 272*cos(c + d*x)*sin(c + d*x)**2 + 345*cos( 
c + d*x)*sin(c + d*x) - 544*cos(c + d*x) - 345*d*x + 544)/(240*a**3*d)