\(\int (1+2 \cos ^9(x))^{5/6} \tan (x) \, dx\) [449]

Optimal result

Integrand size = 15, antiderivative size = 95 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {\arctan \left (\frac {1-\sqrt [3]{1+2 \cos ^9(x)}}{\sqrt {3} \sqrt [6]{1+2 \cos ^9(x)}}\right )}{3 \sqrt {3}}+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{9} \text {arctanh}\left (\sqrt {1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6} \] Output:

1/3*arctanh((1+2*cos(x)^9)^(1/6))-1/9*arctanh((1+2*cos(x)^9)^(1/2))-2/15*( 
1+2*cos(x)^9)^(5/6)+1/9*arctan(1/3*(1-(1+2*cos(x)^9)^(1/3))/(1+2*cos(x)^9) 
^(1/6)*3^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.62 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {1}{90} \left (10 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )-10 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )+20 \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-12 \left (1+2 \cos ^9(x)\right )^{5/6}-5 \log \left (1-\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )+5 \log \left (1+\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )\right ) \] Input:

Integrate[(1 + 2*Cos[x]^9)^(5/6)*Tan[x],x]
 

Output:

(10*Sqrt[3]*ArcTan[(1 - 2*(1 + 2*Cos[x]^9)^(1/6))/Sqrt[3]] - 10*Sqrt[3]*Ar 
cTan[(1 + 2*(1 + 2*Cos[x]^9)^(1/6))/Sqrt[3]] + 20*ArcTanh[(1 + 2*Cos[x]^9) 
^(1/6)] - 12*(1 + 2*Cos[x]^9)^(5/6) - 5*Log[1 - (1 + 2*Cos[x]^9)^(1/6) + ( 
1 + 2*Cos[x]^9)^(1/3)] + 5*Log[1 + (1 + 2*Cos[x]^9)^(1/6) + (1 + 2*Cos[x]^ 
9)^(1/3)])/90
 

Rubi [A] (warning: unable to verify)

Time = 0.35 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.67, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 3709, 798, 60, 73, 27, 825, 27, 219, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \left (2 \cos ^9(x)+1\right )^{5/6} \tan (x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\left (2 \sin \left (x+\frac {\pi }{2}\right )^9+1\right )^{5/6}}{\tan \left (x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\left (2 \sin \left (x+\frac {\pi }{2}\right )^9+1\right )^{5/6}}{\tan \left (x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3709

\(\displaystyle -\int \left (2 \cos ^9(x)+1\right )^{5/6} \sec (x)d\cos (x)\)

\(\Big \downarrow \) 798

\(\displaystyle -\frac {1}{9} \int \left (2 \cos ^9(x)+1\right )^{5/6} \sec (x)d\cos ^9(x)\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{9} \left (-\int \frac {\sec (x)}{\sqrt [6]{2 \cos ^9(x)+1}}d\cos ^9(x)-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{9} \left (-3 \int -\frac {2 \cos ^{36}(x)}{1-\cos ^{54}(x)}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (6 \int \frac {\cos ^{36}(x)}{1-\cos ^{54}(x)}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 825

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{3} \int \frac {1}{1-\cos ^{18}(x)}d\sqrt [6]{2 \cos ^9(x)+1}+\frac {1}{3} \int -\frac {\sqrt [6]{2 \cos ^9(x)+1}+1}{2 \left (\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1\right )}d\sqrt [6]{2 \cos ^9(x)+1}+\frac {1}{3} \int -\frac {1-\sqrt [6]{2 \cos ^9(x)+1}}{2 \left (\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1\right )}d\sqrt [6]{2 \cos ^9(x)+1}\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{3} \int \frac {1}{1-\cos ^{18}(x)}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {1}{6} \int \frac {\sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {1}{6} \int \frac {1-\sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{9} \left (6 \left (-\frac {1}{6} \int \frac {\sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {1}{6} \int \frac {1-\sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {1}{2} \int -\frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {3}{2} \int \frac {1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {3}{2} \int \frac {1}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\frac {3}{2} \int \frac {1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{6} \left (3 \int \frac {1}{-\cos ^{18}(x)-3}d\left (2 \sqrt [6]{2 \cos ^9(x)+1}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\cos ^{18}(x)-3}d\left (2 \sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac {1}{2} \int \frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1}d\sqrt [6]{2 \cos ^9(x)+1}-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{9} \left (6 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\cos ^{18}(x)-\sqrt [6]{2 \cos ^9(x)+1}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (\cos ^{18}(x)+\sqrt [6]{2 \cos ^9(x)+1}+1\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right )-\frac {6}{5} \left (2 \cos ^9(x)+1\right )^{5/6}\right )\)

Input:

Int[(1 + 2*Cos[x]^9)^(5/6)*Tan[x],x]
 

Output:

((-6*(1 + 2*Cos[x]^9)^(5/6))/5 + 6*(ArcTanh[(1 + 2*Cos[x]^9)^(1/6)]/3 + (- 
(Sqrt[3]*ArcTan[(-1 + 2*(1 + 2*Cos[x]^9)^(1/6))/Sqrt[3]]) - Log[1 + Cos[x] 
^18 - (1 + 2*Cos[x]^9)^(1/6)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + 2*(1 + 2*Cos[x 
]^9)^(1/6))/Sqrt[3]]) + Log[1 + Cos[x]^18 + (1 + 2*Cos[x]^9)^(1/6)]/2)/6)) 
/9
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 825

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k 
*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + 
s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 
 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))   Int[1/ 
(r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], 
x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 
] && NegQ[a/b]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 3042

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

rule 3709

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si 
mp[ff^(m + 1)/f   Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 
1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && 
ILtQ[(m - 1)/2, 0]
 

Maple [F]

\[\int \left (1+2 \cos \left (x \right )^{9}\right )^{\frac {5}{6}} \tan \left (x \right )d x\]

Input:

int((1+2*cos(x)^9)^(5/6)*tan(x),x)
 

Output:

int((1+2*cos(x)^9)^(5/6)*tan(x),x)
 

Fricas [F(-1)]

Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\text {Timed out} \] Input:

integrate((1+2*cos(x)^9)^(5/6)*tan(x),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\text {Timed out} \] Input:

integrate((1+2*cos(x)**9)**(5/6)*tan(x),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (72) = 144\).

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.53 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right ) \] Input:

integrate((1+2*cos(x)^9)^(5/6)*tan(x),x, algorithm="maxima")
 

Output:

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) + 1)) - 1/9*sqrt 
(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) - 1)) - 2/15*(2*cos(x)^9 
+ 1)^(5/6) + 1/18*log((2*cos(x)^9 + 1)^(1/3) + (2*cos(x)^9 + 1)^(1/6) + 1) 
 - 1/18*log((2*cos(x)^9 + 1)^(1/3) - (2*cos(x)^9 + 1)^(1/6) + 1) + 1/9*log 
((2*cos(x)^9 + 1)^(1/6) + 1) - 1/9*log((2*cos(x)^9 + 1)^(1/6) - 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (72) = 144\).

Time = 0.17 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.54 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left | {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \] Input:

integrate((1+2*cos(x)^9)^(5/6)*tan(x),x, algorithm="giac")
 

Output:

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) + 1)) - 1/9*sqrt 
(3)*arctan(1/3*sqrt(3)*(2*(2*cos(x)^9 + 1)^(1/6) - 1)) - 2/15*(2*cos(x)^9 
+ 1)^(5/6) + 1/18*log((2*cos(x)^9 + 1)^(1/3) + (2*cos(x)^9 + 1)^(1/6) + 1) 
 - 1/18*log((2*cos(x)^9 + 1)^(1/3) - (2*cos(x)^9 + 1)^(1/6) + 1) + 1/9*log 
((2*cos(x)^9 + 1)^(1/6) + 1) - 1/9*log(abs((2*cos(x)^9 + 1)^(1/6) - 1))
 

Mupad [F(-1)]

Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\int \mathrm {tan}\left (x\right )\,{\left (2\,{\cos \left (x\right )}^9+1\right )}^{5/6} \,d x \] Input:

int(tan(x)*(2*cos(x)^9 + 1)^(5/6),x)
 

Output:

int(tan(x)*(2*cos(x)^9 + 1)^(5/6), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.20 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {\left (2 \cos \left (x \right )^{9}+1\right )^{\frac {5}{6}} \mathrm {log}\left (\tan \left (x \right )^{2}+1\right )}{2} \] Input:

int((1+2*cos(x)^9)^(5/6)*tan(x),x)
 

Output:

((2*cos(x)**9 + 1)**(5/6)*log(tan(x)**2 + 1))/2