\(\int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [741]

Optimal result

Integrand size = 27, antiderivative size = 131 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 x}{16 a^3}-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )} \]

[Out]

-7/16*x/a^3-7/30*cos(d*x+c)^5/a^3/d-7/16*cos(d*x+c)*sin(d*x+c)/a^3/d-7/24*cos(d*x+c)^3*sin(d*x+c)/a^3/d-1/3*co
s(d*x+c)^9/d/(a+a*sin(d*x+c))^3-1/6*cos(d*x+c)^7/d/(a^3+a^3*sin(d*x+c))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2938, 2758, 2761, 2715, 8} \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {\cos ^7(c+d x)}{6 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac {7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {7 x}{16 a^3}-\frac {\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3} \]

[In]

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

[Out]

(-7*x)/(16*a^3) - (7*Cos[c + d*x]^5)/(30*a^3*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (7*Cos[c + d*x]^3
*Sin[c + d*x])/(24*a^3*d) - Cos[c + d*x]^9/(3*d*(a + a*Sin[c + d*x])^3) - Cos[c + d*x]^7/(6*d*(a^3 + a^3*Sin[c
+ d*x]))

Rule 8

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

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 2758

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(a*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p,
0] && IntegersQ[2*m, 2*p]

Rule 2761

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

Rule 2938

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p
+ 1))), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a} \\ & = -\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \frac {\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx}{6 a^2} \\ & = -\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \cos ^4(c+d x) \, dx}{6 a^3} \\ & = -\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int \cos ^2(c+d x) \, dx}{8 a^3} \\ & = -\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {7 \int 1 \, dx}{16 a^3} \\ & = -\frac {7 x}{16 a^3}-\frac {7 \cos ^5(c+d x)}{30 a^3 d}-\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^9(c+d x)}{3 d (a+a \sin (c+d x))^3}-\frac {\cos ^7(c+d x)}{6 d \left (a^3+a^3 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(366\) vs. \(2(131)=262\).

Time = 1.73 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.79 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-21 (1+40 d x) \cos \left (\frac {c}{2}\right )-600 \cos \left (\frac {c}{2}+d x\right )-600 \cos \left (\frac {3 c}{2}+d x\right )+15 \cos \left (\frac {3 c}{2}+2 d x\right )-15 \cos \left (\frac {5 c}{2}+2 d x\right )-140 \cos \left (\frac {5 c}{2}+3 d x\right )-140 \cos \left (\frac {7 c}{2}+3 d x\right )+105 \cos \left (\frac {7 c}{2}+4 d x\right )-105 \cos \left (\frac {9 c}{2}+4 d x\right )+36 \cos \left (\frac {9 c}{2}+5 d x\right )+36 \cos \left (\frac {11 c}{2}+5 d x\right )-5 \cos \left (\frac {11 c}{2}+6 d x\right )+5 \cos \left (\frac {13 c}{2}+6 d x\right )+21 \sin \left (\frac {c}{2}\right )-840 d x \sin \left (\frac {c}{2}\right )+600 \sin \left (\frac {c}{2}+d x\right )-600 \sin \left (\frac {3 c}{2}+d x\right )+15 \sin \left (\frac {3 c}{2}+2 d x\right )+15 \sin \left (\frac {5 c}{2}+2 d x\right )+140 \sin \left (\frac {5 c}{2}+3 d x\right )-140 \sin \left (\frac {7 c}{2}+3 d x\right )+105 \sin \left (\frac {7 c}{2}+4 d x\right )+105 \sin \left (\frac {9 c}{2}+4 d x\right )-36 \sin \left (\frac {9 c}{2}+5 d x\right )+36 \sin \left (\frac {11 c}{2}+5 d x\right )-5 \sin \left (\frac {11 c}{2}+6 d x\right )-5 \sin \left (\frac {13 c}{2}+6 d x\right )}{1920 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]

[In]

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

[Out]

(-21*(1 + 40*d*x)*Cos[c/2] - 600*Cos[c/2 + d*x] - 600*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5
*c)/2 + 2*d*x] - 140*Cos[(5*c)/2 + 3*d*x] - 140*Cos[(7*c)/2 + 3*d*x] + 105*Cos[(7*c)/2 + 4*d*x] - 105*Cos[(9*c
)/2 + 4*d*x] + 36*Cos[(9*c)/2 + 5*d*x] + 36*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 +
6*d*x] + 21*Sin[c/2] - 840*d*x*Sin[c/2] + 600*Sin[c/2 + d*x] - 600*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*
x] + 15*Sin[(5*c)/2 + 2*d*x] + 140*Sin[(5*c)/2 + 3*d*x] - 140*Sin[(7*c)/2 + 3*d*x] + 105*Sin[(7*c)/2 + 4*d*x]
+ 105*Sin[(9*c)/2 + 4*d*x] - 36*Sin[(9*c)/2 + 5*d*x] + 36*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*
Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60

method	result	size
parallelrisch	\(\frac {-420 d x -600 \cos \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )+36 \cos \left (5 d x +5 c \right )+105 \sin \left (4 d x +4 c \right )-140 \cos \left (3 d x +3 c \right )+15 \sin \left (2 d x +2 c \right )-704}{960 d \,a^{3}}\)	\(78\)
risch	\(-\frac {7 x}{16 a^{3}}-\frac {5 \cos \left (d x +c \right )}{8 a^{3} d}-\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{3}}+\frac {3 \cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {7 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}-\frac {7 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {\sin \left (2 d x +2 c \right )}{64 d \,a^{3}}\)	\(107\)
derivativedivides	\(\frac {\frac {4 \left (-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {73 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {9 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {37 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {73 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\)	\(181\)
default	\(\frac {\frac {4 \left (-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {73 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {9 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {37 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {11 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {37 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {73 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {17 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}-\frac {11}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{3}}\)	\(181\)

[In]

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

[Out]

1/960*(-420*d*x-600*cos(d*x+c)-5*sin(6*d*x+6*c)+36*cos(5*d*x+5*c)+105*sin(4*d*x+4*c)-140*cos(3*d*x+3*c)+15*sin
(2*d*x+2*c)-704)/d/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {144 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 50 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{3} d} \]

[In]

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

[Out]

1/240*(144*cos(d*x + c)^5 - 320*cos(d*x + c)^3 - 105*d*x - 5*(8*cos(d*x + c)^5 - 50*cos(d*x + c)^3 + 21*cos(d*
x + c))*sin(d*x + c))/(a^3*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2535 vs. \(2 (121) = 242\).

Time = 130.26 (sec) , antiderivative size = 2535, normalized size of antiderivative = 19.35 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-105*d*x*tan(c/2 + d*x/2)**12/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 +
3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a*
*3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 630*d*x*tan(c/2 + d*x/2)**10/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*
a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*
tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 1575*d*x*tan(c/2 + d*x/2)**8/(240*a**3*d
*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c
/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 2100*d*x*ta
n(c/2 + d*x/2)**6/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 +
d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**
2 + 240*a**3*d) - 1575*d*x*tan(c/2 + d*x/2)**4/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)
**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1
440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 630*d*x*tan(c/2 + d*x/2)**2/(240*a**3*d*tan(c/2 + d*x/2)**12 +
1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a*
*3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 105*d*x/(240*a**3*d*tan(c/2 + d*x/2
)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 +
3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 210*tan(c/2 + d*x/2)**11/(2
40*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**
3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 48
0*tan(c/2 + d*x/2)**10/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c
/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x
/2)**2 + 240*a**3*d) + 730*tan(c/2 + d*x/2)**9/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)
**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1
440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 4320*tan(c/2 + d*x/2)**8/(240*a**3*d*tan(c/2 + d*x/2)**12 + 144
0*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*
d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) + 2220*tan(c/2 + d*x/2)**7/(240*a**3*d*t
an(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2
+ d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 3520*tan(c/2
+ d*x/2)**6/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)
**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 24
0*a**3*d) - 2220*tan(c/2 + d*x/2)**5/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 360
0*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d
*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 960*tan(c/2 + d*x/2)**4/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*ta
n(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 +
d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 730*tan(c/2 + d*x/2)**3/(240*a**3*d*tan(c/2 + d*x
/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6
+ 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) - 1632*tan(c/2 + d*x/2)**2/
(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2 + d*x/2)**8 + 4800*a
**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2)**2 + 240*a**3*d) +
210*tan(c/2 + d*x/2)/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan(c/2
+ d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d*x/2
)**2 + 240*a**3*d) - 352/(240*a**3*d*tan(c/2 + d*x/2)**12 + 1440*a**3*d*tan(c/2 + d*x/2)**10 + 3600*a**3*d*tan
(c/2 + d*x/2)**8 + 4800*a**3*d*tan(c/2 + d*x/2)**6 + 3600*a**3*d*tan(c/2 + d*x/2)**4 + 1440*a**3*d*tan(c/2 + d
*x/2)**2 + 240*a**3*d), Ne(d, 0)), (x*sin(c)*cos(c)**8/(a*sin(c) + a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (119) = 238\).

Time = 0.30 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.00 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {816 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {365 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1110 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1760 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1110 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2160 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {365 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {240 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 176}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \]

[In]

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

[Out]

1/120*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 816*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 365*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 - 480*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 1110*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1760*
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1110*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 2160*sin(d*x + c)^8/(cos(d*x
+ c) + 1)^8 + 365*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 240*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 105*sin(d*
x + c)^11/(cos(d*x + c) + 1)^11 - 176)/(a^3 + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x + c)^
4/(cos(d*x + c) + 1)^4 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)
^8 + 6*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 105*arctan(sin
(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {105 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1760 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 816 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 176\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \]

[In]

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

[Out]

-1/240*(105*(d*x + c)/a^3 + 2*(105*tan(1/2*d*x + 1/2*c)^11 + 240*tan(1/2*d*x + 1/2*c)^10 - 365*tan(1/2*d*x + 1
/2*c)^9 + 2160*tan(1/2*d*x + 1/2*c)^8 - 1110*tan(1/2*d*x + 1/2*c)^7 + 1760*tan(1/2*d*x + 1/2*c)^6 + 1110*tan(1
/2*d*x + 1/2*c)^5 + 480*tan(1/2*d*x + 1/2*c)^4 + 365*tan(1/2*d*x + 1/2*c)^3 + 816*tan(1/2*d*x + 1/2*c)^2 - 105
*tan(1/2*d*x + 1/2*c) + 176)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^3))/d

Mupad [B] (verification not implemented)

Time = 13.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7\,x}{16\,a^3}-\frac {\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {22}{15}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]

[In]

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

[Out]

- (7*x)/(16*a^3) - ((34*tan(c/2 + (d*x)/2)^2)/5 - (7*tan(c/2 + (d*x)/2))/8 + (73*tan(c/2 + (d*x)/2)^3)/24 + 4*
tan(c/2 + (d*x)/2)^4 + (37*tan(c/2 + (d*x)/2)^5)/4 + (44*tan(c/2 + (d*x)/2)^6)/3 - (37*tan(c/2 + (d*x)/2)^7)/4
+ 18*tan(c/2 + (d*x)/2)^8 - (73*tan(c/2 + (d*x)/2)^9)/24 + 2*tan(c/2 + (d*x)/2)^10 + (7*tan(c/2 + (d*x)/2)^11
)/8 + 22/15)/(a^3*d*(tan(c/2 + (d*x)/2)^2 + 1)^6)