\(\int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx\) [84]

Optimal result

Integrand size = 21, antiderivative size = 168 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {661 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]

[Out]

x/a^5-1/9*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-13/63*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4-34
/105*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+173/315*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-661/315*sin(
d*x+c)/d/(a^5+a^5*cos(d*x+c))

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2844, 3056, 3047, 3098, 2814, 2727} \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {661 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {x}{a^5}+\frac {173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

[In]

Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5,x]

[Out]

x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*
(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (173*Sin[c + d
*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (661*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2844

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

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3098

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_
.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a*B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + D
ist[1/(a^2*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^3(c+d x) (4 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^2(c+d x) \left (39 a^2-63 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (204 a^3-315 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {\int \frac {204 a^3 \cos (c+d x)-315 a^3 \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {-1038 a^4+945 a^4 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8} \\ & = \frac {x}{a^5}-\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {661 \int \frac {1}{a+a \cos (c+d x)} \, dx}{315 a^4} \\ & = \frac {x}{a^5}-\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {661 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {8 \cos \left (\frac {1}{2} (c+d x)\right ) \csc ^{10}(c+d x) \sin ^{11}\left (\frac {1}{2} (c+d x)\right ) \left (80640 \arcsin (\cos (c+d x)) \cos ^{10}\left (\frac {1}{2} (c+d x)\right )+(20689+33440 \cos (c+d x)+17648 \cos (2 (c+d x))+5480 \cos (3 (c+d x))+863 \cos (4 (c+d x))) \sqrt {\sin ^2(c+d x)}\right )}{315 a^5 d \sqrt {\sin ^2(c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5,x]

[Out]

(-8*Cos[(c + d*x)/2]*Csc[c + d*x]^10*Sin[(c + d*x)/2]^11*(80640*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^10 + (20
689 + 33440*Cos[c + d*x] + 17648*Cos[2*(c + d*x)] + 5480*Cos[3*(c + d*x)] + 863*Cos[4*(c + d*x)])*Sqrt[Sin[c +
d*x]^2]))/(315*a^5*d*Sqrt[Sin[c + d*x]^2])

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46

method	result	size
parallelrisch	\(\frac {-35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+270 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2730 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5040 d x -9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 a^{5} d}\)	\(77\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(85\)
default	\(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(85\)
risch	\(\frac {x}{a^{5}}-\frac {2 i \left (1575 \,{\mathrm e}^{8 i \left (d x +c \right )}+9450 \,{\mathrm e}^{7 i \left (d x +c \right )}+28350 \,{\mathrm e}^{6 i \left (d x +c \right )}+50400 \,{\mathrm e}^{5 i \left (d x +c \right )}+58338 \,{\mathrm e}^{4 i \left (d x +c \right )}+44142 \,{\mathrm e}^{3 i \left (d x +c \right )}+21618 \,{\mathrm e}^{2 i \left (d x +c \right )}+6192 \,{\mathrm e}^{i \left (d x +c \right )}+863\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\)	\(119\)

[In]

int(cos(d*x+c)^5/(a+cos(d*x+c)*a)^5,x,method=_RETURNVERBOSE)

[Out]

1/5040*(-35*tan(1/2*d*x+1/2*c)^9+270*tan(1/2*d*x+1/2*c)^7-1008*tan(1/2*d*x+1/2*c)^5+2730*tan(1/2*d*x+1/2*c)^3+
5040*d*x-9765*tan(1/2*d*x+1/2*c))/a^5/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/315*(315*d*x*cos(d*x + c)^5 + 1575*d*x*cos(d*x + c)^4 + 3150*d*x*cos(d*x + c)^3 + 3150*d*x*cos(d*x + c)^2 +
1575*d*x*cos(d*x + c) + 315*d*x - (863*cos(d*x + c)^4 + 2740*cos(d*x + c)^3 + 3549*cos(d*x + c)^2 + 2125*cos(d
*x + c) + 488)*sin(d*x + c))/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5
*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)

Sympy [A] (verification not implemented)

Time = 7.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {x}{a^{5}} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {3 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5 a^{5} d} + \frac {13 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} - \frac {31 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5/(a+a*cos(d*x+c))**5,x)

[Out]

Piecewise((x/a**5 - tan(c/2 + d*x/2)**9/(144*a**5*d) + 3*tan(c/2 + d*x/2)**7/(56*a**5*d) - tan(c/2 + d*x/2)**5
/(5*a**5*d) + 13*tan(c/2 + d*x/2)**3/(24*a**5*d) - 31*tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x*cos(c)**5/(a
*cos(c) + a)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/5040*((9765*sin(d*x + c)/(cos(d*x + c) + 1) - 2730*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1008*sin(d*x + c)^
5/(cos(d*x + c) + 1)^5 - 270*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5
- 10080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(5040*(d*x + c)/a^5 - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 270*a^40*tan(1/2*d*x + 1/2*c)^7 + 1008*a^40*tan
(1/2*d*x + 1/2*c)^5 - 2730*a^40*tan(1/2*d*x + 1/2*c)^3 + 9765*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

Mupad [B] (verification not implemented)

Time = 14.85 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

[In]

int(cos(c + d*x)^5/(a + a*cos(c + d*x))^5,x)

[Out]

x/a^5 - (sin(c/2 + (d*x)/2)/144 - (41*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2))/504 + (169*cos(c/2 + (d*x)/2)^4
*sin(c/2 + (d*x)/2))/420 - (356*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2))/315 + (863*cos(c/2 + (d*x)/2)^8*sin(c
/2 + (d*x)/2))/315)/(a^5*d*cos(c/2 + (d*x)/2)^9)