\(\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [708]
Optimal result
Integrand size = 29, antiderivative size = 139 \[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 x}{128 a}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {5 \cos (c+d x) \sin (c+d x)}{128 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{48 a d}-\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}
\]
[Out]
5/128*x/a+1/7*cos(d*x+c)^7/a/d-1/9*cos(d*x+c)^9/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
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2648, 2715, 8, 2645, 14}
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^9(c+d x)}{9 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{48 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}+\frac {5 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac {5 x}{128 a}
\]
[In]
Int[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
(5*x)/(128*a) + Cos[c + d*x]^7/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (5
*Cos[c + d*x]^3*Sin[c + d*x])/(192*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) - (Cos[c + d*x]^7*Sin[c + d*x
])/(8*a*d)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 2645
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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 2648
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] + Dist[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 2715
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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]
Rule 2918
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[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]
Rubi steps \begin{align*}
\text {integral}& = \frac {\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^3(c+d x) \, dx}{a} \\ & = -\frac {\cos ^7(c+d x) \sin (c+d x)}{8 a d}+\frac {\int \cos ^6(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{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}+\frac {5 \int \cos ^4(c+d x) \, dx}{48 a}+\frac {\text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 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}+\frac {5 \int \cos ^2(c+d x) \, dx}{64 a} \\ & = \frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 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}+\frac {5 \int 1 \, dx}{128 a} \\ & = \frac {5 x}{128 a}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 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} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(479\) vs. \(2(139)=278\).
Time = 6.76 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.45
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2520 (c-2 d x) \cos \left (\frac {c}{2}\right )-1512 \cos \left (\frac {c}{2}+d x\right )-1512 \cos \left (\frac {3 c}{2}+d x\right )-1008 \cos \left (\frac {3 c}{2}+2 d x\right )+1008 \cos \left (\frac {5 c}{2}+2 d x\right )-672 \cos \left (\frac {5 c}{2}+3 d x\right )-672 \cos \left (\frac {7 c}{2}+3 d x\right )+504 \cos \left (\frac {7 c}{2}+4 d x\right )-504 \cos \left (\frac {9 c}{2}+4 d x\right )+336 \cos \left (\frac {11 c}{2}+6 d x\right )-336 \cos \left (\frac {13 c}{2}+6 d x\right )+108 \cos \left (\frac {13 c}{2}+7 d x\right )+108 \cos \left (\frac {15 c}{2}+7 d x\right )+63 \cos \left (\frac {15 c}{2}+8 d x\right )-63 \cos \left (\frac {17 c}{2}+8 d x\right )+28 \cos \left (\frac {17 c}{2}+9 d x\right )+28 \cos \left (\frac {19 c}{2}+9 d x\right )-7560 \sin \left (\frac {c}{2}\right )+2520 c \sin \left (\frac {c}{2}\right )-5040 d x \sin \left (\frac {c}{2}\right )+1512 \sin \left (\frac {c}{2}+d x\right )-1512 \sin \left (\frac {3 c}{2}+d x\right )-1008 \sin \left (\frac {3 c}{2}+2 d x\right )-1008 \sin \left (\frac {5 c}{2}+2 d x\right )+672 \sin \left (\frac {5 c}{2}+3 d x\right )-672 \sin \left (\frac {7 c}{2}+3 d x\right )+504 \sin \left (\frac {7 c}{2}+4 d x\right )+504 \sin \left (\frac {9 c}{2}+4 d x\right )+336 \sin \left (\frac {11 c}{2}+6 d x\right )+336 \sin \left (\frac {13 c}{2}+6 d x\right )-108 \sin \left (\frac {13 c}{2}+7 d x\right )+108 \sin \left (\frac {15 c}{2}+7 d x\right )+63 \sin \left (\frac {15 c}{2}+8 d x\right )+63 \sin \left (\frac {17 c}{2}+8 d x\right )-28 \sin \left (\frac {17 c}{2}+9 d x\right )+28 \sin \left (\frac {19 c}{2}+9 d x\right )}{129024 a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )}
\]
[In]
Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
-1/129024*(2520*(c - 2*d*x)*Cos[c/2] - 1512*Cos[c/2 + d*x] - 1512*Cos[(3*c)/2 + d*x] - 1008*Cos[(3*c)/2 + 2*d*
x] + 1008*Cos[(5*c)/2 + 2*d*x] - 672*Cos[(5*c)/2 + 3*d*x] - 672*Cos[(7*c)/2 + 3*d*x] + 504*Cos[(7*c)/2 + 4*d*x
] - 504*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(11*c)/2 + 6*d*x] - 336*Cos[(13*c)/2 + 6*d*x] + 108*Cos[(13*c)/2 + 7*d*
x] + 108*Cos[(15*c)/2 + 7*d*x] + 63*Cos[(15*c)/2 + 8*d*x] - 63*Cos[(17*c)/2 + 8*d*x] + 28*Cos[(17*c)/2 + 9*d*x
] + 28*Cos[(19*c)/2 + 9*d*x] - 7560*Sin[c/2] + 2520*c*Sin[c/2] - 5040*d*x*Sin[c/2] + 1512*Sin[c/2 + d*x] - 151
2*Sin[(3*c)/2 + d*x] - 1008*Sin[(3*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 2*d*x] + 672*Sin[(5*c)/2 + 3*d*x] - 672*
Sin[(7*c)/2 + 3*d*x] + 504*Sin[(7*c)/2 + 4*d*x] + 504*Sin[(9*c)/2 + 4*d*x] + 336*Sin[(11*c)/2 + 6*d*x] + 336*S
in[(13*c)/2 + 6*d*x] - 108*Sin[(13*c)/2 + 7*d*x] + 108*Sin[(15*c)/2 + 7*d*x] + 63*Sin[(15*c)/2 + 8*d*x] + 63*S
in[(17*c)/2 + 8*d*x] - 28*Sin[(17*c)/2 + 9*d*x] + 28*Sin[(19*c)/2 + 9*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))
Maple [A] (verified)
Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.72
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {2520 d x -108 \cos \left (7 d x +7 c \right )+672 \cos \left (3 d x +3 c \right )+1512 \cos \left (d x +c \right )-28 \cos \left (9 d x +9 c \right )-63 \sin \left (8 d x +8 c \right )-336 \sin \left (6 d x +6 c \right )-504 \sin \left (4 d x +4 c \right )+1008 \sin \left (2 d x +2 c \right )+2048}{64512 d a}\) |
\(100\) |
| risch |
\(\frac {5 x}{128 a}+\frac {3 \cos \left (d x +c \right )}{128 a d}-\frac {\cos \left (9 d x +9 c \right )}{2304 a d}-\frac {\sin \left (8 d x +8 c \right )}{1024 d a}-\frac {3 \cos \left (7 d x +7 c \right )}{1792 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}-\frac {\sin \left (4 d x +4 c \right )}{128 d a}+\frac {\cos \left (3 d x +3 c \right )}{96 a d}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\) |
\(141\) |
| derivativedivides |
\(\frac {\frac {8 \left (\frac {1}{126}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {191 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {83 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {145 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {145 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {83 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {191 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {5 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) |
\(233\) |
| default |
\(\frac {\frac {8 \left (\frac {1}{126}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {191 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {83 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {145 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {145 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {83 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {191 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {5 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d a}\) |
\(233\) |
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[In]
int(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/64512*(2520*d*x-108*cos(7*d*x+7*c)+672*cos(3*d*x+3*c)+1512*cos(d*x+c)-28*cos(9*d*x+9*c)-63*sin(8*d*x+8*c)-33
6*sin(6*d*x+6*c)-504*sin(4*d*x+4*c)+1008*sin(2*d*x+2*c)+2048)/d/a
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {896 \, \cos \left (d x + c\right )^{9} - 1152 \, \cos \left (d x + c\right )^{7} - 315 \, d x + 21 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, a d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-1/8064*(896*cos(d*x + c)^9 - 1152*cos(d*x + c)^7 - 315*d*x + 21*(48*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*co
s(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. 4490 vs. \(2 (116) = 232\).
Time = 77.71 (sec) , antiderivative size = 4490, normalized size of antiderivative = 32.30
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**8*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((315*d*x*tan(c/2 + d*x/2)**18/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 2903
04*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d
*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d
*x/2)**2 + 8064*a*d) + 2835*d*x*tan(c/2 + d*x/2)**16/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/
2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10
+ 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*
d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 11340*d*x*tan(c/2 + d*x/2)**14/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*
tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2
+ d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)
**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 26460*d*x*tan(c/2 + d*x/2)**12/(8064*a*d*tan(c/2 + d*x/2)**1
8 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 10160
64*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*ta
n(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 39690*d*x*tan(c/2 + d*x/2)**10/(8064*a*d*tan(c
/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/
2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 +
290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 39690*d*x*tan(c/2 + d*x/2)**8/(8
064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*t
an(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2
+ d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 26460*d*x*tan(c/2 +
d*x/2)**6/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 +
677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376
*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 11340*
d*x*tan(c/2 + d*x/2)**4/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 +
d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2
)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*
a*d) + 2835*d*x*tan(c/2 + d*x/2)**2/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a
*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan
(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2
)**2 + 8064*a*d) + 315*d*x/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/
2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*
x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 80
64*a*d) + 630*tan(c/2 + d*x/2)**17/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*
d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(
c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)
**2 + 8064*a*d) - 16044*tan(c/2 + d*x/2)**15/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 +
290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 101606
4*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/
2 + d*x/2)**2 + 8064*a*d) + 32256*tan(c/2 + d*x/2)**14/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*
x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**1
0 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*
a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 20916*tan(c/2 + d*x/2)**13/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*ta
n(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 +
d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**
4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) - 53760*tan(c/2 + d*x/2)**12/(8064*a*d*tan(c/2 + d*x/2)**18 + 72
576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d
*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2
+ d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) - 36540*tan(c/2 + d*x/2)**11/(8064*a*d*tan(c/2 + d*x/2
)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1
016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*
d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 161280*tan(c/2 + d*x/2)**10/(8064*a*d*tan(
c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x
/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6
+ 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) - 96768*tan(c/2 + d*x/2)**8/(8064
*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(
c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d
*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 36540*tan(c/2 + d*x/2)
**7/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376
*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*ta
n(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) + 96768*tan(c/2
+ d*x/2)**6/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d*x/2)**14
+ 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)**8 + 6773
76*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*d) - 2091
6*tan(c/2 + d*x/2)**5/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*tan(c/2 + d
*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2 + d*x/2)*
*8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2 + 8064*a*
d) - 13824*tan(c/2 + d*x/2)**4/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 290304*a*d*ta
n(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*tan(c/2
+ d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*x/2)**2
+ 8064*a*d) + 16044*tan(c/2 + d*x/2)**3/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 2903
04*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d
*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d
*x/2)**2 + 8064*a*d) + 4608*tan(c/2 + d*x/2)**2/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**1
6 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 101
6064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan
(c/2 + d*x/2)**2 + 8064*a*d) - 630*tan(c/2 + d*x/2)/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2
)**16 + 290304*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 +
1016064*a*d*tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d
*tan(c/2 + d*x/2)**2 + 8064*a*d) + 512/(8064*a*d*tan(c/2 + d*x/2)**18 + 72576*a*d*tan(c/2 + d*x/2)**16 + 29030
4*a*d*tan(c/2 + d*x/2)**14 + 677376*a*d*tan(c/2 + d*x/2)**12 + 1016064*a*d*tan(c/2 + d*x/2)**10 + 1016064*a*d*
tan(c/2 + d*x/2)**8 + 677376*a*d*tan(c/2 + d*x/2)**6 + 290304*a*d*tan(c/2 + d*x/2)**4 + 72576*a*d*tan(c/2 + d*
x/2)**2 + 8064*a*d), Ne(d, 0)), (x*sin(c)**2*cos(c)**8/(a*sin(c) + a), True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (125) = 250\).
Time = 0.33 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.76
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2304 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8022 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6912 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10458 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {48384 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {18270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {48384 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {80640 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {18270 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {26880 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {10458 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {16128 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {8022 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {315 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - 256}{a + \frac {9 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {36 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {126 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {84 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {36 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {9 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}}} - \frac {315 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{4032 \, d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/4032*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 2304*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8022*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 + 6912*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10458*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 -
48384*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 18270*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 48384*sin(d*x + c)^8/
(cos(d*x + c) + 1)^8 - 80640*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 18270*sin(d*x + c)^11/(cos(d*x + c) + 1)^
11 + 26880*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 10458*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 16128*sin(d*x
+ c)^14/(cos(d*x + c) + 1)^14 + 8022*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 315*sin(d*x + c)^17/(cos(d*x + c
) + 1)^17 - 256)/(a + 9*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 36*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 84*
a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 126*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*a*sin(d*x + c)^10/(cos
(d*x + c) + 1)^10 + 84*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 36*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 +
9*a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18) - 315*arctan(sin(d*x + c)
/(cos(d*x + c) + 1))/a)/d
Giac [A] (verification not implemented)
none
Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.66
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {315 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 8022 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 16128 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 10458 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 26880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 18270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 80640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 48384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10458 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8022 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2304 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{9} a}}{8064 \, d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
1/8064*(315*(d*x + c)/a + 2*(315*tan(1/2*d*x + 1/2*c)^17 - 8022*tan(1/2*d*x + 1/2*c)^15 + 16128*tan(1/2*d*x +
1/2*c)^14 + 10458*tan(1/2*d*x + 1/2*c)^13 - 26880*tan(1/2*d*x + 1/2*c)^12 - 18270*tan(1/2*d*x + 1/2*c)^11 + 80
640*tan(1/2*d*x + 1/2*c)^10 - 48384*tan(1/2*d*x + 1/2*c)^8 + 18270*tan(1/2*d*x + 1/2*c)^7 + 48384*tan(1/2*d*x
+ 1/2*c)^6 - 10458*tan(1/2*d*x + 1/2*c)^5 - 6912*tan(1/2*d*x + 1/2*c)^4 + 8022*tan(1/2*d*x + 1/2*c)^3 + 2304*t
an(1/2*d*x + 1/2*c)^2 - 315*tan(1/2*d*x + 1/2*c) + 256)/((tan(1/2*d*x + 1/2*c)^2 + 1)^9*a))/d
Mupad [B] (verification not implemented)
Time = 13.43 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.61
\[
\int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,x}{128\,a}+\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {145\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {145\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}-\frac {12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}+\frac {191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {4}{63}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9}
\]
[In]
int((cos(c + d*x)^8*sin(c + d*x)^2)/(a + a*sin(c + d*x)),x)
[Out]
(5*x)/(128*a) + ((4*tan(c/2 + (d*x)/2)^2)/7 - (5*tan(c/2 + (d*x)/2))/64 + (191*tan(c/2 + (d*x)/2)^3)/96 - (12*
tan(c/2 + (d*x)/2)^4)/7 - (83*tan(c/2 + (d*x)/2)^5)/32 + 12*tan(c/2 + (d*x)/2)^6 + (145*tan(c/2 + (d*x)/2)^7)/
32 - 12*tan(c/2 + (d*x)/2)^8 + 20*tan(c/2 + (d*x)/2)^10 - (145*tan(c/2 + (d*x)/2)^11)/32 - (20*tan(c/2 + (d*x)
/2)^12)/3 + (83*tan(c/2 + (d*x)/2)^13)/32 + 4*tan(c/2 + (d*x)/2)^14 - (191*tan(c/2 + (d*x)/2)^15)/96 + (5*tan(
c/2 + (d*x)/2)^17)/64 + 4/63)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^9)