\(\int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx\) [1246]

Optimal result

Integrand size = 13, antiderivative size = 90 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {1}{18} \log \left (-1+\sqrt [3]{1+x^6}\right )+\frac {1}{36} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \] Output:

-1/6*(x^6+1)^(2/3)/x^6-1/18*arctan(1/3*3^(1/2)+2/3*(x^6+1)^(1/3)*3^(1/2))* 
3^(1/2)-1/18*ln(-1+(x^6+1)^(1/3))+1/36*ln(1+(x^6+1)^(1/3)+(x^6+1)^(2/3))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=\frac {1}{36} \left (-\frac {6 \left (1+x^6\right )^{2/3}}{x^6}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{1+x^6}\right )+\log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \] Input:

Integrate[1/(x^7*(1 + x^6)^(1/3)),x]
 

Output:

((-6*(1 + x^6)^(2/3))/x^6 - 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)])/36
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 52, 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.

\(\displaystyle \int \frac {1}{x^7 \sqrt [3]{x^6+1}} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{6} \int \frac {1}{x^{12} \sqrt [3]{x^6+1}}dx^6\)

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

Input:

Int[1/(x^7*(1 + x^6)^(1/3)),x]
 

Output:

(-((1 + x^6)^(2/3)/x^6) + (-(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)/3)/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 52

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

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 = 4.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{36 \pi }\)	\(76\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {4 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], -x^{6}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi }\)	\(77\)
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{6}+\ln \left (1+\left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right ) x^{6}-2 \ln \left (-1+\left (x^{6}+1\right )^{\frac {1}{3}}\right ) x^{6}-6 \left (x^{6}+1\right )^{\frac {2}{3}}}{36 \left (1+\left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right ) \left (-1+\left (x^{6}+1\right )^{\frac {1}{3}}\right )}\)	\(103\)
trager	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}-\frac {\ln \left (\frac {738615580 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{6}-1815598906 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{6}[phone] x^{6}[phone] \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-738615580 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}-2061146520 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}-769555624 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-569431707 \left (x^{6}+1\right )^{\frac {1}{3}}[phone]}{x^{6}}\right )}{18}+\frac {\ln \left (\frac {676735492 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{6}-30940044 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{6}-707675536 x^{6}-3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-676735492 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-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 (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )[phone] \left (x^{6}+1\right )^{\frac {1}{3}}-1769188840}{x^{6}}\right )}{18}-\frac {\ln \left (\frac {676735492 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{6}-30940044 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{6}-707675536 x^{6}-3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-676735492 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-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 (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )[phone] \left (x^{6}+1\right )^{\frac {1}{3}}-1769188840}{x^{6}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{9}\)	\(427\)

Input:

int(1/x^7/(x^6+1)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

-1/6*(x^6+1)^(2/3)/x^6-1/36/Pi*3^(1/2)*GAMMA(2/3)*(-2/9*Pi*3^(1/2)/GAMMA(2 
/3)*x^6*hypergeom([1,1,4/3],[2,2],-x^6)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+6*l 
n(x))*Pi*3^(1/2)/GAMMA(2/3))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=-\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) + 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, x^{6}} \] Input:

integrate(1/x^7/(x^6+1)^(1/3),x, algorithm="fricas")
 

Output:

-1/36*(2*sqrt(3)*x^6*arctan(2/3*sqrt(3)*(x^6 + 1)^(1/3) + 1/3*sqrt(3)) - x 
^6*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) + 2*x^6*log((x^6 + 1)^(1/3) 
- 1) + 6*(x^6 + 1)^(2/3))/x^6
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=- \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{8} \Gamma \left (\frac {7}{3}\right )} \] Input:

integrate(1/x**7/(x**6+1)**(1/3),x)
 

Output:

-gamma(4/3)*hyper((1/3, 4/3), (7/3,), exp_polar(I*pi)/x**6)/(6*x**8*gamma( 
7/3))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate(1/x^7/(x^6+1)^(1/3),x, algorithm="maxima")
 

Output:

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) - 1/6*(x^6 + 1)^ 
(2/3)/x^6 + 1/36*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) - 1/18*log((x^ 
6 + 1)^(1/3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=-\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{36} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:

integrate(1/x^7/(x^6+1)^(1/3),x, algorithm="giac")
 

Output:

-1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 + 1)^(1/3) + 1)) - 1/6*(x^6 + 1)^ 
(2/3)/x^6 + 1/36*log((x^6 + 1)^(2/3) + (x^6 + 1)^(1/3) + 1) - 1/18*log((x^ 
6 + 1)^(1/3) - 1)
 

Mupad [B] (verification not implemented)

Time = 9.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^7 \sqrt [3]{1+x^6}} \, dx=-\frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-\frac {1}{36}\right )}{18}-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-9\,{\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\right )\,\left (-\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{36}-9\,{\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )}^2\right )\,\left (\frac {1}{36}+\frac {\sqrt {3}\,1{}\mathrm {i}}{36}\right )-\frac {{\left (x^6+1\right )}^{2/3}}{6\,x^6} \] Input:

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

Output:

log((x^6 + 1)^(1/3)/36 - 9*((3^(1/2)*1i)/36 + 1/36)^2)*((3^(1/2)*1i)/36 + 
1/36) - log((x^6 + 1)^(1/3)/36 - 9*((3^(1/2)*1i)/36 - 1/36)^2)*((3^(1/2)*1 
i)/36 - 1/36) - log((x^6 + 1)^(1/3)/36 - 1/36)/18 - (x^6 + 1)^(2/3)/(6*x^6 
)
 

Reduce [F]

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

int(1/x^7/(x^6+1)^(1/3),x)
 

Output:

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