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

Optimal result

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

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))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1009\) vs. \(2(168)=336\).

Time = 10.60 (sec) , antiderivative size = 1009, normalized size of antiderivative = 6.01 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-((Sin[c + d*x]*(-152/(315*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d*x])^(9/ 
2)) - (25*Cos[c + d*x])/(72*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d*x])^(9 
/2)) + (76*Cos[c + d*x]^2)/(35*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d*x]) 
^(9/2)) + (25*Cos[c + d*x]^3)/(24*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d* 
x])^(9/2)) - (19*Cos[c + d*x]^4)/(5*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + 
d*x])^(9/2)) + (80*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^4)/(9*(1 + Cos[c + 
d*x])^(9/2)) - (5*Cos[c + d*x]^5)/(4*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + 
 d*x])^(9/2)) + (16*Cos[c + d*x]^5)/(Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d 
*x])^(9/2)) + (10*Cos[c + d*x]^6)/(3*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + 
 d*x])^(9/2)) - (8*Cos[c + d*x]^6)/(3*(1 - Cos[c + d*x])^(3/2)*(1 + Cos[c 
+ d*x])^(9/2)) + (8*Cos[c + d*x]^7)/(7*(1 - Cos[c + d*x])^(5/2)*(1 + Cos[c 
 + d*x])^(9/2)) - (5*Cos[c + d*x]^8)/(7*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[ 
c + d*x])^(9/2)) - (5*Cos[c + d*x]^9)/(9*(1 - Cos[c + d*x])^(9/2)*(1 + Cos 
[c + d*x])^(9/2)) + 25/(36*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d*x])^(7/ 
2)) + (25*Cos[c + d*x])/(63*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7 
/2)) + (320*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^3)/(63*(1 + Cos[c + d*x])^ 
(7/2)) - 25/(72*(1 - Cos[c + d*x])^(9/2)*(1 + Cos[c + d*x])^(5/2)) - 25/(7 
2*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(5/2)) + (64*Sqrt[1 - Cos[c 
+ d*x]])/(21*(1 + Cos[c + d*x])^(5/2)) + (5*Cos[c + d*x])/(84*(1 - Cos[c + 
 d*x])^(5/2)*(1 + Cos[c + d*x])^(5/2)) - (512*Sqrt[1 - Cos[c + d*x]])/(...
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3244, 3042, 3456, 27, 3042, 3456, 3042, 3447, 3042, 3498, 25, 3042, 3214, 3042, 3127}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\)

\(\Big \downarrow \) 3244

\(\displaystyle -\frac {\int \frac {\cos ^3(c+d x) (4 a-9 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 a-9 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3456

\(\displaystyle -\frac {\frac {\int \frac {3 \cos ^2(c+d x) \left (13 a^2-21 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (13 a^2-21 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (13 a^2-21 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3456

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\cos (c+d x) \left (68 a^3-105 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (68 a^3-105 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3447

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {68 a^3 \cos (c+d x)-105 a^3 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {68 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )-105 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3498

\(\displaystyle -\frac {\frac {3 \left (\frac {-\frac {\int -\frac {346 a^4-315 a^4 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \frac {346 a^4-315 a^4 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \frac {346 a^4-315 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3214

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {661 a^4 \int \frac {1}{\cos (c+d x) a+a}dx-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {661 a^4 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

\(\Big \downarrow \) 3127

\(\displaystyle -\frac {\frac {3 \left (\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {\frac {\frac {661 a^4 \sin (c+d x)}{d (a \cos (c+d x)+a)}-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\)

Input:

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

Output:

-1/9*(Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - ((13*a*Cos 
[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (3*((34*a^2*Cos[c 
 + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((-173*a^3*Sin[c + 
d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (-315*a^3*x + (661*a^4*Sin[c + d*x])/ 
(d*(a + a*Cos[c + d*x])))/(3*a^2))/(5*a^2)))/(7*a^2))/(9*a^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3127

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 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(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 3244

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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] + Simp[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 3447

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 3456

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(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] - Simp[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] /; Fre 
eQ[{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] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

rule 3498

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] + Simp[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]
 

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46

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

Input:

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

Output:

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)

Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \] Input:

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

Output:

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 = 6.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {x}{a^{5}} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {3 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5 a^{5} d} + \frac {13 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} - \frac {31 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \] Input:

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

Output:

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*c 
os(c) + a)**5, True))
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \] Input:

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

Output:

-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)

Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \] Input:

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

Output:

1/5040*(5040*(d*x + c)/a^5 - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 270*a^40*ta 
n(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 = 41.45 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:

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

Output:

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)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.45 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {-35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+270 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-1008 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5040 d x}{5040 a^{5} d} \] Input:

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

Output:

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