\(\int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\) [709]

Optimal result

Integrand size = 27, antiderivative size = 121 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 x}{128 a}-\frac {\cos ^7(c+d x)}{7 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d} \] Output:

-5/128*x/a-1/7*cos(d*x+c)^7/a/d-5/128*cos(d*x+c)*sin(d*x+c)/a/d-5/192*cos( 
d*x+c)^3*sin(d*x+c)/a/d-1/48*cos(d*x+c)^5*sin(d*x+c)/a/d+1/8*cos(d*x+c)^7* 
sin(d*x+c)/a/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(481\) vs. \(2(121)=242\).

Time = 12.83 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.98 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-336 (7 c-5 d x) \cos \left (\frac {c}{2}\right )+1680 \cos \left (\frac {c}{2}+d x\right )+1680 \cos \left (\frac {3 c}{2}+d x\right )+336 \cos \left (\frac {3 c}{2}+2 d x\right )-336 \cos \left (\frac {5 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+3 d x\right )+1008 \cos \left (\frac {7 c}{2}+3 d x\right )-168 \cos \left (\frac {7 c}{2}+4 d x\right )+168 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {9 c}{2}+5 d x\right )+336 \cos \left (\frac {11 c}{2}+5 d x\right )-112 \cos \left (\frac {11 c}{2}+6 d x\right )+112 \cos \left (\frac {13 c}{2}+6 d x\right )+48 \cos \left (\frac {13 c}{2}+7 d x\right )+48 \cos \left (\frac {15 c}{2}+7 d x\right )-21 \cos \left (\frac {15 c}{2}+8 d x\right )+21 \cos \left (\frac {17 c}{2}+8 d x\right )+4704 \sin \left (\frac {c}{2}\right )-2352 c \sin \left (\frac {c}{2}\right )+1680 d x \sin \left (\frac {c}{2}\right )-1680 \sin \left (\frac {c}{2}+d x\right )+1680 \sin \left (\frac {3 c}{2}+d x\right )+336 \sin \left (\frac {3 c}{2}+2 d x\right )+336 \sin \left (\frac {5 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+3 d x\right )+1008 \sin \left (\frac {7 c}{2}+3 d x\right )-168 \sin \left (\frac {7 c}{2}+4 d x\right )-168 \sin \left (\frac {9 c}{2}+4 d x\right )-336 \sin \left (\frac {9 c}{2}+5 d x\right )+336 \sin \left (\frac {11 c}{2}+5 d x\right )-112 \sin \left (\frac {11 c}{2}+6 d x\right )-112 \sin \left (\frac {13 c}{2}+6 d x\right )-48 \sin \left (\frac {13 c}{2}+7 d x\right )+48 \sin \left (\frac {15 c}{2}+7 d x\right )-21 \sin \left (\frac {15 c}{2}+8 d x\right )-21 \sin \left (\frac {17 c}{2}+8 d x\right )}{43008 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:

Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
 

Output:

-1/43008*(-336*(7*c - 5*d*x)*Cos[c/2] + 1680*Cos[c/2 + d*x] + 1680*Cos[(3* 
c)/2 + d*x] + 336*Cos[(3*c)/2 + 2*d*x] - 336*Cos[(5*c)/2 + 2*d*x] + 1008*C 
os[(5*c)/2 + 3*d*x] + 1008*Cos[(7*c)/2 + 3*d*x] - 168*Cos[(7*c)/2 + 4*d*x] 
 + 168*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(9*c)/2 + 5*d*x] + 336*Cos[(11*c)/2 
+ 5*d*x] - 112*Cos[(11*c)/2 + 6*d*x] + 112*Cos[(13*c)/2 + 6*d*x] + 48*Cos[ 
(13*c)/2 + 7*d*x] + 48*Cos[(15*c)/2 + 7*d*x] - 21*Cos[(15*c)/2 + 8*d*x] + 
21*Cos[(17*c)/2 + 8*d*x] + 4704*Sin[c/2] - 2352*c*Sin[c/2] + 1680*d*x*Sin[ 
c/2] - 1680*Sin[c/2 + d*x] + 1680*Sin[(3*c)/2 + d*x] + 336*Sin[(3*c)/2 + 2 
*d*x] + 336*Sin[(5*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 3*d*x] + 1008*Sin[(7 
*c)/2 + 3*d*x] - 168*Sin[(7*c)/2 + 4*d*x] - 168*Sin[(9*c)/2 + 4*d*x] - 336 
*Sin[(9*c)/2 + 5*d*x] + 336*Sin[(11*c)/2 + 5*d*x] - 112*Sin[(11*c)/2 + 6*d 
*x] - 112*Sin[(13*c)/2 + 6*d*x] - 48*Sin[(13*c)/2 + 7*d*x] + 48*Sin[(15*c) 
/2 + 7*d*x] - 21*Sin[(15*c)/2 + 8*d*x] - 21*Sin[(17*c)/2 + 8*d*x])/(a*d*(C 
os[c/2] + Sin[c/2]))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3318, 3042, 3045, 15, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}

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.

\(\displaystyle \int \frac {\sin (c+d x) \cos ^8(c+d x)}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^8}{a \sin (c+d x)+a}dx\)

\(\Big \downarrow \) 3318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3045

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\int \cos (c+d x)^6 \sin (c+d x)^2dx}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3048

\(\displaystyle -\frac {\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}-\frac {\cos ^7(c+d x)}{7 a d}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\cos ^7(c+d x)}{7 a d}-\frac {\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}}{a}\)

Input:

Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
 

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3045

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ 
Symbol] :> Simp[-(a*f)^(-1)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], 
x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && 
 !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
 

rule 3048

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 
1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n))   Int[(b*Cos[e + f*x])^n 
*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] 
 && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3318

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a   Int 
[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d)   Int 
[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, 
d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83

method	result	size
parallelrisch	\(\frac {-840 d x +112 \sin \left (6 d x +6 c \right )+168 \sin \left (4 d x +4 c \right )-336 \sin \left (2 d x +2 c \right )+21 \sin \left (8 d x +8 c \right )-48 \cos \left (7 d x +7 c \right )-336 \cos \left (5 d x +5 c \right )-1008 \cos \left (3 d x +3 c \right )-1680 \cos \left (d x +c \right )-3072}{21504 d a}\)	\(100\)
risch	\(-\frac {5 x}{128 a}-\frac {5 \cos \left (d x +c \right )}{64 a d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\cos \left (5 d x +5 c \right )}{64 a d}+\frac {\sin \left (4 d x +4 c \right )}{128 d a}-\frac {3 \cos \left (3 d x +3 c \right )}{64 a d}-\frac {\sin \left (2 d x +2 c \right )}{64 d a}\)	\(141\)
derivativedivides	\(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}-\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2}+\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\)	\(233\)
default	\(\frac {\frac {4 \left (-\frac {1}{14}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{14}-\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{768}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {1765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{768}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}-\frac {895 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2}+\frac {397 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{768}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{256}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\)	\(233\)

Input:

int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/21504*(-840*d*x+112*sin(6*d*x+6*c)+168*sin(4*d*x+4*c)-336*sin(2*d*x+2*c) 
+21*sin(8*d*x+8*c)-48*cos(7*d*x+7*c)-336*cos(5*d*x+5*c)-1008*cos(3*d*x+3*c 
)-1680*cos(d*x+c)-3072)/d/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {384 \, \cos \left (d x + c\right )^{7} + 105 \, d x - 7 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, a d} \] Input:

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

Output:

-1/2688*(384*cos(d*x + c)^7 + 105*d*x - 7*(48*cos(d*x + c)^7 - 8*cos(d*x + 
 c)^5 - 10*cos(d*x + c)^3 - 15*cos(d*x + c))*sin(d*x + c))/(a*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3888 vs. \(2 (102) = 204\).

Time = 52.32 (sec) , antiderivative size = 3888, normalized size of antiderivative = 32.13 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c)),x)
 

Output:

Piecewise((-105*d*x*tan(c/2 + d*x/2)**16/(2688*a*d*tan(c/2 + d*x/2)**16 + 
21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 150528*a 
*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528*a*d*tan( 
c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/2 + d*x/ 
2)**2 + 2688*a*d) - 840*d*x*tan(c/2 + d*x/2)**14/(2688*a*d*tan(c/2 + d*x/2 
)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/2)**12 + 
150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 + 150528* 
a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a*d*tan(c/ 
2 + d*x/2)**2 + 2688*a*d) - 2940*d*x*tan(c/2 + d*x/2)**12/(2688*a*d*tan(c/ 
2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c/2 + d*x/ 
2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d*x/2)**8 
+ 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 + 21504*a 
*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 5880*d*x*tan(c/2 + d*x/2)**10/(2688*a 
*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a*d*tan(c 
/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan(c/2 + d 
*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 + d*x/2)**4 
+ 21504*a*d*tan(c/2 + d*x/2)**2 + 2688*a*d) - 7350*d*x*tan(c/2 + d*x/2)**8 
/(2688*a*d*tan(c/2 + d*x/2)**16 + 21504*a*d*tan(c/2 + d*x/2)**14 + 75264*a 
*d*tan(c/2 + d*x/2)**12 + 150528*a*d*tan(c/2 + d*x/2)**10 + 188160*a*d*tan 
(c/2 + d*x/2)**8 + 150528*a*d*tan(c/2 + d*x/2)**6 + 75264*a*d*tan(c/2 +...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (109) = 218\).

Time = 0.15 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.14 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

Output:

1/1344*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x 
 + c) + 1)^2 - 2779*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 8064*sin(d*x + c 
)^4/(cos(d*x + c) + 1)^4 + 6265*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 8064 
*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 12355*sin(d*x + c)^7/(cos(d*x + c) 
+ 1)^7 - 13440*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 12355*sin(d*x + c)^9/ 
(cos(d*x + c) + 1)^9 - 13440*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6265* 
sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 2688*sin(d*x + c)^12/(cos(d*x + c) 
 + 1)^12 + 2779*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 2688*sin(d*x + c)^ 
14/(cos(d*x + c) + 1)^14 - 105*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 384 
)/(a + 8*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a*sin(d*x + c)^4/(cos( 
d*x + c) + 1)^4 + 56*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 70*a*sin(d*x 
+ c)^8/(cos(d*x + c) + 1)^8 + 56*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 
 28*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a*sin(d*x + c)^14/(cos(d*x 
 + c) + 1)^14 + a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16) - 105*arctan(sin( 
d*x + c)/(cos(d*x + c) + 1))/a)/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).

Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.91 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2688 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 12355 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2779 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \] Input:

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

-1/2688*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 2688*tan(1/2*d 
*x + 1/2*c)^14 - 2779*tan(1/2*d*x + 1/2*c)^13 + 2688*tan(1/2*d*x + 1/2*c)^ 
12 + 6265*tan(1/2*d*x + 1/2*c)^11 + 13440*tan(1/2*d*x + 1/2*c)^10 - 12355* 
tan(1/2*d*x + 1/2*c)^9 + 13440*tan(1/2*d*x + 1/2*c)^8 + 12355*tan(1/2*d*x 
+ 1/2*c)^7 + 8064*tan(1/2*d*x + 1/2*c)^6 - 6265*tan(1/2*d*x + 1/2*c)^5 + 8 
064*tan(1/2*d*x + 1/2*c)^4 + 2779*tan(1/2*d*x + 1/2*c)^3 + 384*tan(1/2*d*x 
 + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 384)/((tan(1/2*d*x + 1/2*c)^2 + 1 
)^8*a))/d
 

Mupad [B] (verification not implemented)

Time = 34.74 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,x}{128\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {1765\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {895\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {397\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {2}{7}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \] Input:

int((cos(c + d*x)^8*sin(c + d*x))/(a + a*sin(c + d*x)),x)
 

Output:

- (5*x)/(128*a) - ((2*tan(c/2 + (d*x)/2)^2)/7 - (5*tan(c/2 + (d*x)/2))/64 
+ (397*tan(c/2 + (d*x)/2)^3)/192 + 6*tan(c/2 + (d*x)/2)^4 - (895*tan(c/2 + 
 (d*x)/2)^5)/192 + 6*tan(c/2 + (d*x)/2)^6 + (1765*tan(c/2 + (d*x)/2)^7)/19 
2 + 10*tan(c/2 + (d*x)/2)^8 - (1765*tan(c/2 + (d*x)/2)^9)/192 + 10*tan(c/2 
 + (d*x)/2)^10 + (895*tan(c/2 + (d*x)/2)^11)/192 + 2*tan(c/2 + (d*x)/2)^12 
 - (397*tan(c/2 + (d*x)/2)^13)/192 + 2*tan(c/2 + (d*x)/2)^14 + (5*tan(c/2 
+ (d*x)/2)^15)/64 + 2/7)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^8)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-336 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-1152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-826 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )-384 \cos \left (d x +c \right )-105 d x +384}{2688 a d} \] Input:

int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x)
 

Output:

( - 336*cos(c + d*x)*sin(c + d*x)**7 + 384*cos(c + d*x)*sin(c + d*x)**6 + 
952*cos(c + d*x)*sin(c + d*x)**5 - 1152*cos(c + d*x)*sin(c + d*x)**4 - 826 
*cos(c + d*x)*sin(c + d*x)**3 + 1152*cos(c + d*x)*sin(c + d*x)**2 + 105*co 
s(c + d*x)*sin(c + d*x) - 384*cos(c + d*x) - 105*d*x + 384)/(2688*a*d)