\(\int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [634]

Optimal result

Integrand size = 29, antiderivative size = 135 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8 a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}+\frac {3 \cos ^5(c+d x)}{5 a^2 d}-\frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{3 a^2 d} \] Output:

-1/8*x/a^2-2/3*cos(d*x+c)^3/a^2/d+3/5*cos(d*x+c)^5/a^2/d-1/7*cos(d*x+c)^7/ 
a^2/d-1/8*cos(d*x+c)*sin(d*x+c)/a^2/d+1/4*cos(d*x+c)^3*sin(d*x+c)/a^2/d+1/ 
3*cos(d*x+c)^3*sin(d*x+c)^3/a^2/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(135)=270\).

Time = 3.89 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.10 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {210 (1+8 d x) \cos \left (\frac {c}{2}\right )+1365 \cos \left (\frac {c}{2}+d x\right )+1365 \cos \left (\frac {3 c}{2}+d x\right )-210 \cos \left (\frac {3 c}{2}+2 d x\right )+210 \cos \left (\frac {5 c}{2}+2 d x\right )+175 \cos \left (\frac {5 c}{2}+3 d x\right )+175 \cos \left (\frac {7 c}{2}+3 d x\right )-210 \cos \left (\frac {7 c}{2}+4 d x\right )+210 \cos \left (\frac {9 c}{2}+4 d x\right )-147 \cos \left (\frac {9 c}{2}+5 d x\right )-147 \cos \left (\frac {11 c}{2}+5 d x\right )+70 \cos \left (\frac {11 c}{2}+6 d x\right )-70 \cos \left (\frac {13 c}{2}+6 d x\right )+15 \cos \left (\frac {13 c}{2}+7 d x\right )+15 \cos \left (\frac {15 c}{2}+7 d x\right )-210 \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1365 \sin \left (\frac {c}{2}+d x\right )+1365 \sin \left (\frac {3 c}{2}+d x\right )-210 \sin \left (\frac {3 c}{2}+2 d x\right )-210 \sin \left (\frac {5 c}{2}+2 d x\right )-175 \sin \left (\frac {5 c}{2}+3 d x\right )+175 \sin \left (\frac {7 c}{2}+3 d x\right )-210 \sin \left (\frac {7 c}{2}+4 d x\right )-210 \sin \left (\frac {9 c}{2}+4 d x\right )+147 \sin \left (\frac {9 c}{2}+5 d x\right )-147 \sin \left (\frac {11 c}{2}+5 d x\right )+70 \sin \left (\frac {11 c}{2}+6 d x\right )+70 \sin \left (\frac {13 c}{2}+6 d x\right )-15 \sin \left (\frac {13 c}{2}+7 d x\right )+15 \sin \left (\frac {15 c}{2}+7 d x\right )}{13440 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
 

Output:

-1/13440*(210*(1 + 8*d*x)*Cos[c/2] + 1365*Cos[c/2 + d*x] + 1365*Cos[(3*c)/ 
2 + d*x] - 210*Cos[(3*c)/2 + 2*d*x] + 210*Cos[(5*c)/2 + 2*d*x] + 175*Cos[( 
5*c)/2 + 3*d*x] + 175*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] + 21 
0*Cos[(9*c)/2 + 4*d*x] - 147*Cos[(9*c)/2 + 5*d*x] - 147*Cos[(11*c)/2 + 5*d 
*x] + 70*Cos[(11*c)/2 + 6*d*x] - 70*Cos[(13*c)/2 + 6*d*x] + 15*Cos[(13*c)/ 
2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] - 210*Sin[c/2] + 1680*d*x*Sin[c/2] - 
 1365*Sin[c/2 + d*x] + 1365*Sin[(3*c)/2 + d*x] - 210*Sin[(3*c)/2 + 2*d*x] 
- 210*Sin[(5*c)/2 + 2*d*x] - 175*Sin[(5*c)/2 + 3*d*x] + 175*Sin[(7*c)/2 + 
3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] + 147*Sin[(9* 
c)/2 + 5*d*x] - 147*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70* 
Sin[(13*c)/2 + 6*d*x] - 15*Sin[(13*c)/2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x 
])/(a^2*d*(Cos[c/2] + Sin[c/2]))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 139, 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, 3354, 3042, 3352, 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.

\(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^6(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^6}{(a \sin (c+d x)+a)^2}dx\)

\(\Big \downarrow \) 3354

\(\displaystyle \frac {\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \cos (c+d x)^2 \sin (c+d x)^3 (a-a \sin (c+d x))^2dx}{a^4}\)

\(\Big \downarrow \) 3352

\(\displaystyle \frac {\int \left (a^2 \cos ^2(c+d x) \sin ^5(c+d x)-2 a^2 \cos ^2(c+d x) \sin ^4(c+d x)+a^2 \cos ^2(c+d x) \sin ^3(c+d x)\right )dx}{a^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{3 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^2 x}{8}}{a^4}\)

Input:

Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]
 

Output:

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

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 3352

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig 
[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F 
reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3354

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

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(\frac {-840 d x +147 \cos \left (5 d x +5 c \right )-175 \cos \left (3 d x +3 c \right )-1365 \cos \left (d x +c \right )-15 \cos \left (7 d x +7 c \right )-70 \sin \left (6 d x +6 c \right )+210 \sin \left (4 d x +4 c \right )+210 \sin \left (2 d x +2 c \right )-1408}{6720 d \,a^{2}}\)	\(89\)
risch	\(-\frac {x}{8 a^{2}}-\frac {13 \cos \left (d x +c \right )}{64 a^{2} d}-\frac {\cos \left (7 d x +7 c \right )}{448 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{96 d \,a^{2}}+\frac {7 \cos \left (5 d x +5 c \right )}{320 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{2}}-\frac {5 \cos \left (3 d x +3 c \right )}{192 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{32 d \,a^{2}}\)	\(124\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{48}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}+\frac {97 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{12}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {97 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{10}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{60}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {11}{420}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(181\)
default	\(\frac {\frac {16 \left (-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{48}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}+\frac {97 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{12}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {97 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{10}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{60}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}-\frac {11}{420}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{2}}\)	\(181\)

Input:

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/6720*(-840*d*x+147*cos(5*d*x+5*c)-175*cos(3*d*x+3*c)-1365*cos(d*x+c)-15* 
cos(7*d*x+7*c)-70*sin(6*d*x+6*c)+210*sin(4*d*x+4*c)+210*sin(2*d*x+2*c)-140 
8)/d/a^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {120 \, \cos \left (d x + c\right )^{7} - 504 \, \cos \left (d x + c\right )^{5} + 560 \, \cos \left (d x + c\right )^{3} + 105 \, d x + 35 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, a^{2} d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

-1/840*(120*cos(d*x + c)^7 - 504*cos(d*x + c)^5 + 560*cos(d*x + c)^3 + 105 
*d*x + 35*(8*cos(d*x + c)^5 - 14*cos(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x 
+ c))/(a^2*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3046 vs. \(2 (121) = 242\).

Time = 88.87 (sec) , antiderivative size = 3046, normalized size of antiderivative = 22.56 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+a*sin(d*x+c))**2,x)
 

Output:

Piecewise((-105*d*x*tan(c/2 + d*x/2)**14/(840*a**2*d*tan(c/2 + d*x/2)**14 
+ 5880*a**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 2 
9400*a**2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640 
*a**2*d*tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d 
) - 735*d*x*tan(c/2 + d*x/2)**12/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a 
**2*d*tan(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a** 
2*d*tan(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d* 
tan(c/2 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 2205 
*d*x*tan(c/2 + d*x/2)**10/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*t 
an(c/2 + d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan 
(c/2 + d*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 
 + d*x/2)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*ta 
n(c/2 + d*x/2)**8/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + 
 d*x/2)**12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d 
*x/2)**8 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2 
)**4 + 5880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 3675*d*x*tan(c/2 + 
d*x/2)**6/(840*a**2*d*tan(c/2 + d*x/2)**14 + 5880*a**2*d*tan(c/2 + d*x/2)* 
*12 + 17640*a**2*d*tan(c/2 + d*x/2)**10 + 29400*a**2*d*tan(c/2 + d*x/2)**8 
 + 29400*a**2*d*tan(c/2 + d*x/2)**6 + 17640*a**2*d*tan(c/2 + d*x/2)**4 + 5 
880*a**2*d*tan(c/2 + d*x/2)**2 + 840*a**2*d) - 2205*d*x*tan(c/2 + d*x/2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (121) = 242\).

Time = 0.13 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.08 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1232 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {700 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2016 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3395 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1120 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {7280 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3395 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1680 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {700 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {105 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - 176}{a^{2} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {35 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {21 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {a^{2} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{420 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

1/420*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 1232*sin(d*x + c)^2/(cos(d*x 
 + c) + 1)^2 + 700*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 2016*sin(d*x + c) 
^4/(cos(d*x + c) + 1)^4 - 3395*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1120* 
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 7280*sin(d*x + c)^8/(cos(d*x + c) + 
1)^8 + 3395*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 1680*sin(d*x + c)^10/(co 
s(d*x + c) + 1)^10 - 700*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 105*sin(d 
*x + c)^13/(cos(d*x + c) + 1)^13 - 176)/(a^2 + 7*a^2*sin(d*x + c)^2/(cos(d 
*x + c) + 1)^2 + 21*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 35*a^2*sin(d 
*x + c)^6/(cos(d*x + c) + 1)^6 + 35*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^ 
8 + 21*a^2*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 7*a^2*sin(d*x + c)^12/( 
cos(d*x + c) + 1)^12 + a^2*sin(d*x + c)^14/(cos(d*x + c) + 1)^14) - 105*ar 
ctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3395 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 700 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1232 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7} a^{2}}}{840 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

-1/840*(105*(d*x + c)/a^2 + 2*(105*tan(1/2*d*x + 1/2*c)^13 + 700*tan(1/2*d 
*x + 1/2*c)^11 + 1680*tan(1/2*d*x + 1/2*c)^10 - 3395*tan(1/2*d*x + 1/2*c)^ 
9 + 7280*tan(1/2*d*x + 1/2*c)^8 - 1120*tan(1/2*d*x + 1/2*c)^6 + 3395*tan(1 
/2*d*x + 1/2*c)^5 + 2016*tan(1/2*d*x + 1/2*c)^4 - 700*tan(1/2*d*x + 1/2*c) 
^3 + 1232*tan(1/2*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 176)/((tan(1 
/2*d*x + 1/2*c)^2 + 1)^7*a^2))/d
 

Mupad [B] (verification not implemented)

Time = 35.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{8\,a^2}-\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {97\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {44}{105}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \] Input:

int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + a*sin(c + d*x))^2,x)
 

Output:

- x/(8*a^2) - ((44*tan(c/2 + (d*x)/2)^2)/15 - tan(c/2 + (d*x)/2)/4 - (5*ta 
n(c/2 + (d*x)/2)^3)/3 + (24*tan(c/2 + (d*x)/2)^4)/5 + (97*tan(c/2 + (d*x)/ 
2)^5)/12 - (8*tan(c/2 + (d*x)/2)^6)/3 + (52*tan(c/2 + (d*x)/2)^8)/3 - (97* 
tan(c/2 + (d*x)/2)^9)/12 + 4*tan(c/2 + (d*x)/2)^10 + (5*tan(c/2 + (d*x)/2) 
^11)/3 + tan(c/2 + (d*x)/2)^13/4 + 44/105)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 
1)^7)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-88 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-176 \cos \left (d x +c \right )-105 d x +176}{840 a^{2} d} \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^3/(a+a*sin(d*x+c))^2,x)
 

Output:

(120*cos(c + d*x)*sin(c + d*x)**6 - 280*cos(c + d*x)*sin(c + d*x)**5 + 144 
*cos(c + d*x)*sin(c + d*x)**4 + 70*cos(c + d*x)*sin(c + d*x)**3 - 88*cos(c 
 + d*x)*sin(c + d*x)**2 + 105*cos(c + d*x)*sin(c + d*x) - 176*cos(c + d*x) 
 - 105*d*x + 176)/(840*a**2*d)