\(\int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [739]

Optimal result

Integrand size = 29, antiderivative size = 161 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {29 x}{128 a^3}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d} \]

[Out]

-29/128*x/a^3-4/3*cos(d*x+c)^3/a^3/d+7/5*cos(d*x+c)^5/a^3/d-3/7*cos(d*x+c)^7/a^3/d-29/128*cos(d*x+c)*sin(d*x+c
)/a^3/d+29/64*cos(d*x+c)^3*sin(d*x+c)/a^3/d+29/48*cos(d*x+c)^3*sin(d*x+c)^3/a^3/d+1/8*cos(d*x+c)^3*sin(d*x+c)^
5/a^3/d

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2954, 2952, 2645, 14, 2648, 2715, 8, 276} \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {3 \cos ^7(c+d x)}{7 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac {29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac {29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac {29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac {29 x}{128 a^3} \]

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-29*x)/(128*a^3) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (7*Cos[c + d*x]^5)/(5*a^3*d) - (3*Cos[c + d*x]^7)/(7*a^3*d)
- (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^3*d) + (29*Cos[c + d*x]
^3*Sin[c + d*x]^3)/(48*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a^3*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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 2952

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]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^2(c+d x) \sin ^3(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (a^3 \cos ^2(c+d x) \sin ^3(c+d x)-3 a^3 \cos ^2(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^2(c+d x) \sin ^5(c+d x)-a^3 \cos ^2(c+d x) \sin ^6(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a^3}-\frac {\int \cos ^2(c+d x) \sin ^6(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a^3} \\ & = \frac {\cos ^3(c+d x) \sin ^3(c+d x)}{2 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx}{8 a^3}-\frac {3 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{2 a^3}-\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = \frac {3 \cos ^3(c+d x) \sin (c+d x)}{8 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{16 a^3}-\frac {3 \int \cos ^2(c+d x) \, dx}{8 a^3}-\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = -\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int \cos ^2(c+d x) \, dx}{64 a^3}-\frac {3 \int 1 \, dx}{16 a^3} \\ & = -\frac {3 x}{16 a^3}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d}-\frac {5 \int 1 \, dx}{128 a^3} \\ & = -\frac {29 x}{128 a^3}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {7 \cos ^5(c+d x)}{5 a^3 d}-\frac {3 \cos ^7(c+d x)}{7 a^3 d}-\frac {29 \cos (c+d x) \sin (c+d x)}{128 a^3 d}+\frac {29 \cos ^3(c+d x) \sin (c+d x)}{64 a^3 d}+\frac {29 \cos ^3(c+d x) \sin ^3(c+d x)}{48 a^3 d}+\frac {\cos ^3(c+d x) \sin ^5(c+d x)}{8 a^3 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(482\) vs. \(2(161)=322\).

Time = 3.00 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.99 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {84 (-7+12870 c-580 d x) \cos \left (\frac {c}{2}\right )-38640 \cos \left (\frac {c}{2}+d x\right )-38640 \cos \left (\frac {3 c}{2}+d x\right )+6720 \cos \left (\frac {3 c}{2}+2 d x\right )-6720 \cos \left (\frac {5 c}{2}+2 d x\right )-3920 \cos \left (\frac {5 c}{2}+3 d x\right )-3920 \cos \left (\frac {7 c}{2}+3 d x\right )+5880 \cos \left (\frac {7 c}{2}+4 d x\right )-5880 \cos \left (\frac {9 c}{2}+4 d x\right )+4368 \cos \left (\frac {9 c}{2}+5 d x\right )+4368 \cos \left (\frac {11 c}{2}+5 d x\right )-2240 \cos \left (\frac {11 c}{2}+6 d x\right )+2240 \cos \left (\frac {13 c}{2}+6 d x\right )-720 \cos \left (\frac {13 c}{2}+7 d x\right )-720 \cos \left (\frac {15 c}{2}+7 d x\right )+105 \cos \left (\frac {15 c}{2}+8 d x\right )-105 \cos \left (\frac {17 c}{2}+8 d x\right )-998928 \sin \left (\frac {c}{2}\right )+1081080 c \sin \left (\frac {c}{2}\right )-48720 d x \sin \left (\frac {c}{2}\right )+38640 \sin \left (\frac {c}{2}+d x\right )-38640 \sin \left (\frac {3 c}{2}+d x\right )+6720 \sin \left (\frac {3 c}{2}+2 d x\right )+6720 \sin \left (\frac {5 c}{2}+2 d x\right )+3920 \sin \left (\frac {5 c}{2}+3 d x\right )-3920 \sin \left (\frac {7 c}{2}+3 d x\right )+5880 \sin \left (\frac {7 c}{2}+4 d x\right )+5880 \sin \left (\frac {9 c}{2}+4 d x\right )-4368 \sin \left (\frac {9 c}{2}+5 d x\right )+4368 \sin \left (\frac {11 c}{2}+5 d x\right )-2240 \sin \left (\frac {11 c}{2}+6 d x\right )-2240 \sin \left (\frac {13 c}{2}+6 d x\right )+720 \sin \left (\frac {13 c}{2}+7 d x\right )-720 \sin \left (\frac {15 c}{2}+7 d x\right )+105 \sin \left (\frac {15 c}{2}+8 d x\right )+105 \sin \left (\frac {17 c}{2}+8 d x\right )}{215040 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]

[Out]

(84*(-7 + 12870*c - 580*d*x)*Cos[c/2] - 38640*Cos[c/2 + d*x] - 38640*Cos[(3*c)/2 + d*x] + 6720*Cos[(3*c)/2 + 2
*d*x] - 6720*Cos[(5*c)/2 + 2*d*x] - 3920*Cos[(5*c)/2 + 3*d*x] - 3920*Cos[(7*c)/2 + 3*d*x] + 5880*Cos[(7*c)/2 +
4*d*x] - 5880*Cos[(9*c)/2 + 4*d*x] + 4368*Cos[(9*c)/2 + 5*d*x] + 4368*Cos[(11*c)/2 + 5*d*x] - 2240*Cos[(11*c)
/2 + 6*d*x] + 2240*Cos[(13*c)/2 + 6*d*x] - 720*Cos[(13*c)/2 + 7*d*x] - 720*Cos[(15*c)/2 + 7*d*x] + 105*Cos[(15
*c)/2 + 8*d*x] - 105*Cos[(17*c)/2 + 8*d*x] - 998928*Sin[c/2] + 1081080*c*Sin[c/2] - 48720*d*x*Sin[c/2] + 38640
*Sin[c/2 + d*x] - 38640*Sin[(3*c)/2 + d*x] + 6720*Sin[(3*c)/2 + 2*d*x] + 6720*Sin[(5*c)/2 + 2*d*x] + 3920*Sin[
(5*c)/2 + 3*d*x] - 3920*Sin[(7*c)/2 + 3*d*x] + 5880*Sin[(7*c)/2 + 4*d*x] + 5880*Sin[(9*c)/2 + 4*d*x] - 4368*Si
n[(9*c)/2 + 5*d*x] + 4368*Sin[(11*c)/2 + 5*d*x] - 2240*Sin[(11*c)/2 + 6*d*x] - 2240*Sin[(13*c)/2 + 6*d*x] + 72
0*Sin[(13*c)/2 + 7*d*x] - 720*Sin[(15*c)/2 + 7*d*x] + 105*Sin[(15*c)/2 + 8*d*x] + 105*Sin[(17*c)/2 + 8*d*x])/(
215040*a^3*d*(Cos[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.62

method	result	size
parallelrisch	\(\frac {-24360 d x -720 \cos \left (7 d x +7 c \right )+4368 \cos \left (5 d x +5 c \right )-3920 \cos \left (3 d x +3 c \right )-38640 \cos \left (d x +c \right )+105 \sin \left (8 d x +8 c \right )-2240 \sin \left (6 d x +6 c \right )+5880 \sin \left (4 d x +4 c \right )+6720 \sin \left (2 d x +2 c \right )-38912}{107520 d \,a^{3}}\)	\(100\)
risch	\(-\frac {29 x}{128 a^{3}}-\frac {23 \cos \left (d x +c \right )}{64 a^{3} d}+\frac {\sin \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {3 \cos \left (7 d x +7 c \right )}{448 d \,a^{3}}-\frac {\sin \left (6 d x +6 c \right )}{48 d \,a^{3}}+\frac {13 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}+\frac {7 \sin \left (4 d x +4 c \right )}{128 d \,a^{3}}-\frac {7 \cos \left (3 d x +3 c \right )}{192 d \,a^{3}}+\frac {\sin \left (2 d x +2 c \right )}{16 d \,a^{3}}\)	\(141\)
derivativedivides	\(\frac {\frac {16 \left (-\frac {19}{420}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024}-\frac {38 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105}+\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {61 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}-\frac {1465 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {5117 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {19 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {5117 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {8 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {1465 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {667 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{3}}\)	\(220\)
default	\(\frac {\frac {16 \left (-\frac {19}{420}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024}-\frac {38 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105}+\frac {667 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {61 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60}-\frac {1465 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {5117 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {19 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12}+\frac {5117 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {8 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {1465 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {667 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072}-\frac {29 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {29 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{d \,a^{3}}\)	\(220\)

[In]

int(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/107520*(-24360*d*x-720*cos(7*d*x+7*c)+4368*cos(5*d*x+5*c)-3920*cos(3*d*x+3*c)-38640*cos(d*x+c)+105*sin(8*d*x
+8*c)-2240*sin(6*d*x+6*c)+5880*sin(4*d*x+4*c)+6720*sin(2*d*x+2*c)-38912)/d/a^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.56 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5760 \, \cos \left (d x + c\right )^{7} - 18816 \, \cos \left (d x + c\right )^{5} + 17920 \, \cos \left (d x + c\right )^{3} + 3045 \, d x - 35 \, {\left (48 \, \cos \left (d x + c\right )^{7} - 328 \, \cos \left (d x + c\right )^{5} + 454 \, \cos \left (d x + c\right )^{3} - 87 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, a^{3} d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/13440*(5760*cos(d*x + c)^7 - 18816*cos(d*x + c)^5 + 17920*cos(d*x + c)^3 + 3045*d*x - 35*(48*cos(d*x + c)^7
- 328*cos(d*x + c)^5 + 454*cos(d*x + c)^3 - 87*cos(d*x + c))*sin(d*x + c))/(a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)**3/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (145) = 290\).

Time = 0.43 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.10 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {3045 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {38912 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {23345 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {109312 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {51275 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {179095 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {170240 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {179095 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {286720 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {51275 \, \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 {23345 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {3045 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - 4864}{a^{3} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {70 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {56 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {28 \, a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} + \frac {a^{3} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}}} - \frac {3045 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{6720 \, d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6720*((3045*sin(d*x + c)/(cos(d*x + c) + 1) - 38912*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 23345*sin(d*x + c)
^3/(cos(d*x + c) + 1)^3 - 109312*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 51275*sin(d*x + c)^5/(cos(d*x + c) + 1)
^5 + 14336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 179095*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 170240*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + 179095*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 286720*sin(d*x + c)^10/(cos(d*x + c)
+ 1)^10 + 51275*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 26880*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 23345*s
in(d*x + c)^13/(cos(d*x + c) + 1)^13 - 3045*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 4864)/(a^3 + 8*a^3*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + 28*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 56*a^3*sin(d*x + c)^6/(cos(d*x + c
) + 1)^6 + 70*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 28*a^3*
sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a^3*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + a^3*sin(d*x + c)^16/(cos
(d*x + c) + 1)^16) - 3045*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {3045 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (3045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 23345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 26880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 51275 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 286720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 179095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 170240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 179095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 14336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 51275 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 109312 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 23345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4864\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8} a^{3}}}{13440 \, d} \]

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/13440*(3045*(d*x + c)/a^3 + 2*(3045*tan(1/2*d*x + 1/2*c)^15 + 23345*tan(1/2*d*x + 1/2*c)^13 + 26880*tan(1/2
*d*x + 1/2*c)^12 - 51275*tan(1/2*d*x + 1/2*c)^11 + 286720*tan(1/2*d*x + 1/2*c)^10 - 179095*tan(1/2*d*x + 1/2*c
)^9 + 170240*tan(1/2*d*x + 1/2*c)^8 + 179095*tan(1/2*d*x + 1/2*c)^7 - 14336*tan(1/2*d*x + 1/2*c)^6 + 51275*tan
(1/2*d*x + 1/2*c)^5 + 109312*tan(1/2*d*x + 1/2*c)^4 - 23345*tan(1/2*d*x + 1/2*c)^3 + 38912*tan(1/2*d*x + 1/2*c
)^2 - 3045*tan(1/2*d*x + 1/2*c) + 4864)/((tan(1/2*d*x + 1/2*c)^2 + 1)^8*a^3))/d

Mupad [B] (verification not implemented)

Time = 12.60 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {29\,x}{128\,a^3}-\frac {\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1465\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}-\frac {5117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {76\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {5117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{192}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {1465\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {244\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-\frac {667\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {608\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {29\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {76}{105}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \]

[In]

int((cos(c + d*x)^8*sin(c + d*x)^3)/(a + a*sin(c + d*x))^3,x)

[Out]

- (29*x)/(128*a^3) - ((608*tan(c/2 + (d*x)/2)^2)/105 - (29*tan(c/2 + (d*x)/2))/64 - (667*tan(c/2 + (d*x)/2)^3)
/192 + (244*tan(c/2 + (d*x)/2)^4)/15 + (1465*tan(c/2 + (d*x)/2)^5)/192 - (32*tan(c/2 + (d*x)/2)^6)/15 + (5117*
tan(c/2 + (d*x)/2)^7)/192 + (76*tan(c/2 + (d*x)/2)^8)/3 - (5117*tan(c/2 + (d*x)/2)^9)/192 + (128*tan(c/2 + (d*
x)/2)^10)/3 - (1465*tan(c/2 + (d*x)/2)^11)/192 + 4*tan(c/2 + (d*x)/2)^12 + (667*tan(c/2 + (d*x)/2)^13)/192 + (
29*tan(c/2 + (d*x)/2)^15)/64 + 76/105)/(a^3*d*(tan(c/2 + (d*x)/2)^2 + 1)^8)