\(\int \frac {(1+x^6)^{2/3}}{x} \, dx\) [1190]

Optimal result

Integrand size = 13, antiderivative size = 87 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{4} \left (1+x^6\right )^{2/3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{12} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \] Output:

1/4*(x^6+1)^(2/3)+1/6*arctan(1/3*3^(1/2)+2/3*(x^6+1)^(1/3)*3^(1/2))*3^(1/2 
)+1/6*ln(-1+(x^6+1)^(1/3))-1/12*ln(1+(x^6+1)^(1/3)+(x^6+1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{12} \left (3 \left (1+x^6\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^6}\right )-\log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \] Input:

Integrate[(1 + x^6)^(2/3)/x,x]
 

Output:

(3*(1 + x^6)^(2/3) + 2*Sqrt[3]*ArcTan[(1 + 2*(1 + x^6)^(1/3))/Sqrt[3]] + 2 
*Log[-1 + (1 + x^6)^(1/3)] - Log[1 + (1 + x^6)^(1/3) + (1 + x^6)^(2/3)])/1 
2
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 60, 67, 16, 1083, 217}

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.

\(\displaystyle \int \frac {\left (x^6+1\right )^{2/3}}{x} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{6} \int \frac {\left (x^6+1\right )^{2/3}}{x^6}dx^6\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{6} \left (\int \frac {1}{x^6 \sqrt [3]{x^6+1}}dx^6+\frac {3}{2} \left (x^6+1\right )^{2/3}\right )\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^6+1}}d\sqrt [3]{x^6+1}+\frac {3}{2} \int \frac {1}{x^{12}+\sqrt [3]{x^6+1}+1}d\sqrt [3]{x^6+1}+\frac {3}{2} \left (x^6+1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{6} \left (\frac {3}{2} \int \frac {1}{x^{12}+\sqrt [3]{x^6+1}+1}d\sqrt [3]{x^6+1}+\frac {3}{2} \left (x^6+1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^6+1}\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{6} \left (-3 \int \frac {1}{-x^{12}-3}d\left (2 \sqrt [3]{x^6+1}+1\right )+\frac {3}{2} \left (x^6+1\right )^{2/3}-\frac {1}{2} \log \left (x^6\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^6+1}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )+\frac {3}{2} \left (x^6+1\right )^{2/3}-\frac {\log \left (x^6\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^6+1}\right )\right )\)

Input:

Int[(1 + x^6)^(2/3)/x,x]
 

Output:

((3*(1 + x^6)^(2/3))/2 + Sqrt[3]*ArcTan[(1 + 2*(1 + x^6)^(1/3))/Sqrt[3]] - 
 Log[x^6]/2 + (3*Log[1 - (1 + x^6)^(1/3)])/2)/6
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

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 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 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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 5.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74

method	result	size
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], -x^{6}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi }\)	\(64\)
pseudoelliptic	\(\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4}-\frac {\ln \left (1+\left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (-1+\left (x^{6}+1\right )^{\frac {1}{3}}\right )}{6}\)	\(64\)
trager	\(\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-523021641 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-369307790 x^{6}-1600004967 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}-353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-1600004967 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-569431707 \left (x^{6}+1\right )^{\frac {2}{3}}-1953842735 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-569431707 \left (x^{6}+1\right )^{\frac {1}{3}}-923269475}{x^{6}}\right )}{6}-\frac {\ln \left (-\frac {353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}[phone] x^{6}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}-353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}[phone] \left (x^{6}+1\right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )[phone] \left (x^{6}+1\right )^{\frac {1}{3}}[phone]}{x^{6}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{6}-\frac {\ln \left (-\frac {353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}[phone] x^{6}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}-353837768 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}[phone] \left (x^{6}+1\right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )[phone] \left (x^{6}+1\right )^{\frac {1}{3}}[phone]}{x^{6}}\right )}{6}\)	\(358\)

Input:

int((x^6+1)^(2/3)/x,x,method=_RETURNVERBOSE)
 

Output:

-1/18/Pi*3^(1/2)*GAMMA(2/3)*(-2/3*Pi*3^(1/2)/GAMMA(2/3)*x^6*hypergeom([1/3 
,1,1],[2,2],-x^6)-(3/2-1/6*Pi*3^(1/2)-3/2*ln(3)+6*ln(x))*Pi*3^(1/2)/GAMMA( 
2/3))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate((x^6+1)^(2/3)/x,x, algorithm="fricas")
 

Output:

1/6*sqrt(3)*arctan(2/3*sqrt(3)*(x^6 + 1)^(1/3) + 1/3*sqrt(3)) + 1/4*(x^6 + 
 1)^(2/3) - 1/12*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) + 1/6*log((x^6 
 + 1)^(1/3) - 1)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.41 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=- \frac {x^{4} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {1}{3}\right )} \] Input:

integrate((x**6+1)**(2/3)/x,x)
 

Output:

-x**4*gamma(-2/3)*hyper((-2/3, -2/3), (1/3,), exp_polar(I*pi)/x**6)/(6*gam 
ma(1/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate((x^6+1)^(2/3)/x,x, algorithm="maxima")
 

Output:

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) + 1/4*(x^6 + 1)^(2 
/3) - 1/12*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) + 1/6*log((x^6 + 1)^ 
(1/3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate((x^6+1)^(2/3)/x,x, algorithm="giac")
 

Output:

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) + 1/4*(x^6 + 1)^(2 
/3) - 1/12*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) + 1/6*log((x^6 + 1)^ 
(1/3) - 1)
 

Mupad [B] (verification not implemented)

Time = 8.53 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-\frac {1}{4}\right )}{6}+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (x^6+1\right )}^{2/3}}{4} \] Input:

int((x^6 + 1)^(2/3)/x,x)
 

Output:

log((x^6 + 1)^(1/3)/4 - 1/4)/6 + log((x^6 + 1)^(1/3)/4 - 9*((3^(1/2)*1i)/1 
2 - 1/12)^2)*((3^(1/2)*1i)/12 - 1/12) - log((x^6 + 1)^(1/3)/4 - 9*((3^(1/2 
)*1i)/12 + 1/12)^2)*((3^(1/2)*1i)/12 + 1/12) + (x^6 + 1)^(2/3)/4
 

Reduce [F]

\[ \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx=\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4}+\int \frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{7}+x}d x \] Input:

int((x^6+1)^(2/3)/x,x)
 

Output:

((x**6 + 1)**(2/3) + 4*int((x**6 + 1)**(2/3)/(x**7 + x),x))/4