\(\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [707]
Optimal result
Integrand size = 29, antiderivative size = 165 \[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 x}{256 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}
\]
[Out]
-3/256*x/a-1/7*cos(d*x+c)^7/a/d+1/9*cos(d*x+c)^9/a/d-3/256*cos(d*x+c)*sin(d*x+c)/a/d-1/128*cos(d*x+c)^3*sin(d*
x+c)/a/d-1/160*cos(d*x+c)^5*sin(d*x+c)/a/d+3/80*cos(d*x+c)^7*sin(d*x+c)/a/d+1/10*cos(d*x+c)^7*sin(d*x+c)^3/a/d
Rubi [A] (verified)
Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of
steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2918, 2645, 14, 2648, 2715,
8} \[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac {3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac {3 x}{256 a}
\]
[In]
Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
[Out]
(-3*x)/(256*a) - Cos[c + d*x]^7/(7*a*d) + Cos[c + d*x]^9/(9*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) - (
Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) + (3*Cos[c + d*x]^7*Sin[c + d
*x])/(80*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*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 ^3(c+d x) \, dx}{a}-\frac {\int \cos ^6(c+d x) \sin ^4(c+d x) \, dx}{a} \\ & = \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{10 a}-\frac {\text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^6(c+d x) \, dx}{80 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 {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {\int \cos ^4(c+d x) \, dx}{32 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int \cos ^2(c+d x) \, dx}{128 a} \\ & = -\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d}-\frac {3 \int 1 \, dx}{256 a} \\ & = -\frac {3 x}{256 a}-\frac {\cos ^7(c+d x)}{7 a d}+\frac {\cos ^9(c+d x)}{9 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{256 a d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{128 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{160 a d}+\frac {3 \cos ^7(c+d x) \sin (c+d x)}{80 a d}+\frac {\cos ^7(c+d x) \sin ^3(c+d x)}{10 a d} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(533\) vs. \(2(165)=330\).
Time = 10.03 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.23
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {-1260 (25 c-12 d x) \cos \left (\frac {c}{2}\right )+15120 \cos \left (\frac {c}{2}+d x\right )+15120 \cos \left (\frac {3 c}{2}+d x\right )+1260 \cos \left (\frac {3 c}{2}+2 d x\right )-1260 \cos \left (\frac {5 c}{2}+2 d x\right )+6720 \cos \left (\frac {5 c}{2}+3 d x\right )+6720 \cos \left (\frac {7 c}{2}+3 d x\right )-2520 \cos \left (\frac {7 c}{2}+4 d x\right )+2520 \cos \left (\frac {9 c}{2}+4 d x\right )-630 \cos \left (\frac {11 c}{2}+6 d x\right )+630 \cos \left (\frac {13 c}{2}+6 d x\right )-1080 \cos \left (\frac {13 c}{2}+7 d x\right )-1080 \cos \left (\frac {15 c}{2}+7 d x\right )+315 \cos \left (\frac {15 c}{2}+8 d x\right )-315 \cos \left (\frac {17 c}{2}+8 d x\right )-280 \cos \left (\frac {17 c}{2}+9 d x\right )-280 \cos \left (\frac {19 c}{2}+9 d x\right )+126 \cos \left (\frac {19 c}{2}+10 d x\right )-126 \cos \left (\frac {21 c}{2}+10 d x\right )+37800 \sin \left (\frac {c}{2}\right )-31500 c \sin \left (\frac {c}{2}\right )+15120 d x \sin \left (\frac {c}{2}\right )-15120 \sin \left (\frac {c}{2}+d x\right )+15120 \sin \left (\frac {3 c}{2}+d x\right )+1260 \sin \left (\frac {3 c}{2}+2 d x\right )+1260 \sin \left (\frac {5 c}{2}+2 d x\right )-6720 \sin \left (\frac {5 c}{2}+3 d x\right )+6720 \sin \left (\frac {7 c}{2}+3 d x\right )-2520 \sin \left (\frac {7 c}{2}+4 d x\right )-2520 \sin \left (\frac {9 c}{2}+4 d x\right )-630 \sin \left (\frac {11 c}{2}+6 d x\right )-630 \sin \left (\frac {13 c}{2}+6 d x\right )+1080 \sin \left (\frac {13 c}{2}+7 d x\right )-1080 \sin \left (\frac {15 c}{2}+7 d x\right )+315 \sin \left (\frac {15 c}{2}+8 d x\right )+315 \sin \left (\frac {17 c}{2}+8 d x\right )+280 \sin \left (\frac {17 c}{2}+9 d x\right )-280 \sin \left (\frac {19 c}{2}+9 d x\right )+126 \sin \left (\frac {19 c}{2}+10 d x\right )+126 \sin \left (\frac {21 c}{2}+10 d x\right )}{1290240 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]^3)/(a + a*Sin[c + d*x]),x]
[Out]
-1/1290240*(-1260*(25*c - 12*d*x)*Cos[c/2] + 15120*Cos[c/2 + d*x] + 15120*Cos[(3*c)/2 + d*x] + 1260*Cos[(3*c)/
2 + 2*d*x] - 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c
)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 630*Cos[(11*c)/2 + 6*d*x] + 630*Cos[(13*c)/2 + 6*d*x] - 1080*Cos[(1
3*c)/2 + 7*d*x] - 1080*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] - 280*Cos
[(17*c)/2 + 9*d*x] - 280*Cos[(19*c)/2 + 9*d*x] + 126*Cos[(19*c)/2 + 10*d*x] - 126*Cos[(21*c)/2 + 10*d*x] + 378
00*Sin[c/2] - 31500*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 15120*Sin[c/2 + d*x] + 15120*Sin[(3*c)/2 + d*x] + 1260*S
in[(3*c)/2 + 2*d*x] + 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c)/2 + 3*d*x] + 6720*Sin[(7*c)/2 + 3*d*x] - 2520
*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] - 630*Sin[(11*c)/2 + 6*d*x] - 630*Sin[(13*c)/2 + 6*d*x] + 10
80*Sin[(13*c)/2 + 7*d*x] - 1080*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x]
+ 280*Sin[(17*c)/2 + 9*d*x] - 280*Sin[(19*c)/2 + 9*d*x] + 126*Sin[(19*c)/2 + 10*d*x] + 126*Sin[(21*c)/2 + 10*d
*x])/(a*d*(Cos[c/2] + Sin[c/2]))
Maple [A] (verified)
Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.67
| | |
| method | result | size |
| | |
| parallelrisch |
\(\frac {-7560 d x +1080 \cos \left (7 d x +7 c \right )-6720 \cos \left (3 d x +3 c \right )-15120 \cos \left (d x +c \right )-126 \sin \left (10 d x +10 c \right )+280 \cos \left (9 d x +9 c \right )-315 \sin \left (8 d x +8 c \right )+630 \sin \left (6 d x +6 c \right )+2520 \sin \left (4 d x +4 c \right )-1260 \sin \left (2 d x +2 c \right )-20480}{645120 d a}\) |
\(111\) |
| risch |
\(-\frac {3 x}{256 a}-\frac {3 \cos \left (d x +c \right )}{128 a d}-\frac {\sin \left (10 d x +10 c \right )}{5120 d a}+\frac {\cos \left (9 d x +9 c \right )}{2304 a d}-\frac {\sin \left (8 d x +8 c \right )}{2048 d a}+\frac {3 \cos \left (7 d x +7 c \right )}{1792 a d}+\frac {\sin \left (6 d x +6 c \right )}{1024 d a}+\frac {\sin \left (4 d x +4 c \right )}{256 d a}-\frac {\cos \left (3 d x +3 c \right )}{96 a d}-\frac {\sin \left (2 d x +2 c \right )}{512 d a}\) |
\(158\) |
| derivativedivides |
\(\frac {\frac {16 \left (-\frac {1}{252}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}+\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {867 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {9 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {519 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}-\frac {1879 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1879 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {519 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {867 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {29 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {3 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) |
\(259\) |
| default |
\(\frac {\frac {16 \left (-\frac {1}{252}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2048}-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{126}+\frac {29 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}-\frac {867 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {9 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14}+\frac {519 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}-\frac {1879 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {1879 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {519 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {\left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {867 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2560}-\frac {\left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {29 \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}-\frac {3 \left (\tan ^{19}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2048}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}}{d a}\) |
\(259\) |
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[In]
int(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/645120*(-7560*d*x+1080*cos(7*d*x+7*c)-6720*cos(3*d*x+3*c)-15120*cos(d*x+c)-126*sin(10*d*x+10*c)+280*cos(9*d*
x+9*c)-315*sin(8*d*x+8*c)+630*sin(6*d*x+6*c)+2520*sin(4*d*x+4*c)-1260*sin(2*d*x+2*c)-20480)/d/a
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.55
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8960 \, \cos \left (d x + c\right )^{9} - 11520 \, \cos \left (d x + c\right )^{7} - 945 \, d x - 63 \, {\left (128 \, \cos \left (d x + c\right )^{9} - 176 \, \cos \left (d x + c\right )^{7} + 8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, a d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
1/80640*(8960*cos(d*x + c)^9 - 11520*cos(d*x + c)^7 - 945*d*x - 63*(128*cos(d*x + c)^9 - 176*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. 5501 vs. \(2 (138) = 276\).
Time = 118.03 (sec) , antiderivative size = 5501, normalized size of antiderivative = 33.34
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**8*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((-945*d*x*tan(c/2 + d*x/2)**20/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3
628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321
280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*
d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 9450*d*x*tan(c/2 + d*x/2)**18/(80640*a*d
*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c
/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2
+ d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)
**2 + 80640*a*d) - 42525*d*x*tan(c/2 + d*x/2)**16/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2
)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**1
2 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3
628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 113400*d*x*tan(c/2 + d*x/2)**14
/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 967680
0*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*
a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c
/2 + d*x/2)**2 + 80640*a*d) - 198450*d*x*tan(c/2 + d*x/2)**12/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan
(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2
+ d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d
*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 238140*d*x*tan(c/2
+ d*x/2)**10/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)*
*16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10
+ 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 8064
00*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 198450*d*x*tan(c/2 + d*x/2)**8/(80640*a*d*tan(c/2 + d*x/2)**20 + 806
400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*
a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*
tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 113400*d
*x*tan(c/2 + d*x/2)**6/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2
+ d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 +
d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)
**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 42525*d*x*tan(c/2 + d*x/2)**4/(80640*a*d*tan(c/2 + d*x/2)*
*20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 +
16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 967
6800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) -
9450*d*x*tan(c/2 + d*x/2)**2/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*
tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan
(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 +
d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 945*d*x/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*
d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*ta
n(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/
2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 1890*tan(c/2 +
d*x/2)**19/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**
16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10
+ 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 80640
0*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 18270*tan(c/2 + d*x/2)**17/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a
*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*t
an(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c
/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 322560*tan(c/
2 + d*x/2)**16/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2
)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**
10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 80
6400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) + 436968*tan(c/2 + d*x/2)**15/(80640*a*d*tan(c/2 + d*x/2)**20 + 8064
00*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a
*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*t
an(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) + 215040*ta
n(c/2 + d*x/2)**14/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d
*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/
2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4
+ 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 1307880*tan(c/2 + d*x/2)**13/(80640*a*d*tan(c/2 + d*x/2)**20 +
806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934
400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*
a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 1075
200*tan(c/2 + d*x/2)**12/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c
/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2
+ d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/
2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) + 2367540*tan(c/2 + d*x/2)**11/(80640*a*d*tan(c/2 + d*x/2)
**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 +
16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 96
76800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d)
- 645120*tan(c/2 + d*x/2)**10/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*
tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan
(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 +
d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 2367540*tan(c/2 + d*x/2)**9/(80640*a*d*tan(c/2 + d*
x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**
14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8
+ 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a
*d) + 1307880*tan(c/2 + d*x/2)**7/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*
a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d
*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c
/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 829440*tan(c/2 + d*x/2)**6/(80640*a*d*tan(c/2 +
d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2
)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)*
*8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 8064
0*a*d) - 436968*tan(c/2 + d*x/2)**5/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 362880
0*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a
*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan
(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) + 92160*tan(c/2 + d*x/2)**4/(80640*a*d*tan(c/2
+ d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/
2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)
**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 806
40*a*d) + 18270*tan(c/2 + d*x/2)**3/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 362880
0*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a
*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan
(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 51200*tan(c/2 + d*x/2)**2/(80640*a*d*tan(c/2
+ d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/
2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)
**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 806
40*a*d) + 1890*tan(c/2 + d*x/2)/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d*tan(c/2 + d*x/2)**18 + 3628800*a*
d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan(c/2 + d*x/2)**12 + 20321280*a*d*t
an(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2 + d*x/2)**6 + 3628800*a*d*tan(c/2
+ d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d) - 5120/(80640*a*d*tan(c/2 + d*x/2)**20 + 806400*a*d
*tan(c/2 + d*x/2)**18 + 3628800*a*d*tan(c/2 + d*x/2)**16 + 9676800*a*d*tan(c/2 + d*x/2)**14 + 16934400*a*d*tan
(c/2 + d*x/2)**12 + 20321280*a*d*tan(c/2 + d*x/2)**10 + 16934400*a*d*tan(c/2 + d*x/2)**8 + 9676800*a*d*tan(c/2
+ d*x/2)**6 + 3628800*a*d*tan(c/2 + d*x/2)**4 + 806400*a*d*tan(c/2 + d*x/2)**2 + 80640*a*d), Ne(d, 0)), (x*si
n(c)**3*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. 583 vs. \(2 (149) = 298\).
Time = 0.43 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.53
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25600 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9135 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {46080 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {218484 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {414720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {653940 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {1183770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {322560 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {1183770 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {537600 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {653940 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} + \frac {107520 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {218484 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {161280 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {9135 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {945 \, \sin \left (d x + c\right )^{19}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{19}} - 2560}{a + \frac {10 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {210 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {252 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {210 \, a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {120 \, a \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {45 \, a \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} + \frac {10 \, a \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}} + \frac {a \sin \left (d x + c\right )^{20}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{20}}} - \frac {945 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{40320 \, d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
1/40320*((945*sin(d*x + c)/(cos(d*x + c) + 1) - 25600*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 9135*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 + 46080*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 218484*sin(d*x + c)^5/(cos(d*x + c) + 1)^
5 - 414720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 653940*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 1183770*sin(d*x
+ c)^9/(cos(d*x + c) + 1)^9 - 322560*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1183770*sin(d*x + c)^11/(cos(d*x
+ c) + 1)^11 - 537600*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 653940*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 + 1
07520*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + 218484*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 161280*sin(d*x +
c)^16/(cos(d*x + c) + 1)^16 - 9135*sin(d*x + c)^17/(cos(d*x + c) + 1)^17 - 945*sin(d*x + c)^19/(cos(d*x + c) +
1)^19 - 2560)/(a + 10*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 45*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 120*
a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 210*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 252*a*sin(d*x + c)^10/(cos
(d*x + c) + 1)^10 + 210*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 120*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14
+ 45*a*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 + 10*a*sin(d*x + c)^18/(cos(d*x + c) + 1)^18 + a*sin(d*x + c)^20/
(cos(d*x + c) + 1)^20) - 945*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Giac [A] (verification not implemented)
none
Time = 0.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.56
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {945 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{19} + 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 161280 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 107520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 537600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 322560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1183770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 653940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 414720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 218484 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 46080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2560\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{10} a}}{80640 \, d}
\]
[In]
integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-1/80640*(945*(d*x + c)/a + 2*(945*tan(1/2*d*x + 1/2*c)^19 + 9135*tan(1/2*d*x + 1/2*c)^17 + 161280*tan(1/2*d*x
+ 1/2*c)^16 - 218484*tan(1/2*d*x + 1/2*c)^15 - 107520*tan(1/2*d*x + 1/2*c)^14 + 653940*tan(1/2*d*x + 1/2*c)^1
3 + 537600*tan(1/2*d*x + 1/2*c)^12 - 1183770*tan(1/2*d*x + 1/2*c)^11 + 322560*tan(1/2*d*x + 1/2*c)^10 + 118377
0*tan(1/2*d*x + 1/2*c)^9 - 653940*tan(1/2*d*x + 1/2*c)^7 + 414720*tan(1/2*d*x + 1/2*c)^6 + 218484*tan(1/2*d*x
+ 1/2*c)^5 - 46080*tan(1/2*d*x + 1/2*c)^4 - 9135*tan(1/2*d*x + 1/2*c)^3 + 25600*tan(1/2*d*x + 1/2*c)^2 - 945*t
an(1/2*d*x + 1/2*c) + 2560)/((tan(1/2*d*x + 1/2*c)^2 + 1)^10*a))/d
Mupad [B] (verification not implemented)
Time = 13.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.52
\[
\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3\,x}{256\,a}-\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}+\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}+\frac {867\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {4}{63}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}}
\]
[In]
int((cos(c + d*x)^8*sin(c + d*x)^3)/(a + a*sin(c + d*x)),x)
[Out]
- (3*x)/(256*a) - ((40*tan(c/2 + (d*x)/2)^2)/63 - (3*tan(c/2 + (d*x)/2))/128 - (29*tan(c/2 + (d*x)/2)^3)/128 -
(8*tan(c/2 + (d*x)/2)^4)/7 + (867*tan(c/2 + (d*x)/2)^5)/160 + (72*tan(c/2 + (d*x)/2)^6)/7 - (519*tan(c/2 + (d
*x)/2)^7)/32 + (1879*tan(c/2 + (d*x)/2)^9)/64 + 8*tan(c/2 + (d*x)/2)^10 - (1879*tan(c/2 + (d*x)/2)^11)/64 + (4
0*tan(c/2 + (d*x)/2)^12)/3 + (519*tan(c/2 + (d*x)/2)^13)/32 - (8*tan(c/2 + (d*x)/2)^14)/3 - (867*tan(c/2 + (d*
x)/2)^15)/160 + 4*tan(c/2 + (d*x)/2)^16 + (29*tan(c/2 + (d*x)/2)^17)/128 + (3*tan(c/2 + (d*x)/2)^19)/128 + 4/6
3)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^10)