\(\int \frac {1}{x (-27+2 x^7)^{2/3}} \, dx\) [301]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}-\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right ) \] Output:

-1/18*ln(x)+1/42*ln(3+(2*x^7-27)^(1/3))-1/63*arctan(1/9*(3-2*(2*x^7-27)^(1 
/3))*3^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{126} \left (-2 \sqrt {3} \arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )+2 \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\log \left (9-3 \sqrt [3]{-27+2 x^7}+\left (-27+2 x^7\right )^{2/3}\right )\right ) \] Input:

Integrate[1/(x*(-27 + 2*x^7)^(2/3)),x]
 

Output:

(-2*Sqrt[3]*ArcTan[(3 - 2*(-27 + 2*x^7)^(1/3))/(3*Sqrt[3])] + 2*Log[3 + (- 
27 + 2*x^7)^(1/3)] - Log[9 - 3*(-27 + 2*x^7)^(1/3) + (-27 + 2*x^7)^(2/3)]) 
/126
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 70, 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.

\(\displaystyle \int \frac {1}{x \left (2 x^7-27\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{7} \int \frac {1}{x^7 \left (2 x^7-27\right )^{2/3}}dx^7\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int \frac {1}{\sqrt [3]{2 x^7-27}+3}d\sqrt [3]{2 x^7-27}+\frac {1}{2} \int \frac {1}{x^{14}-3 \sqrt [3]{2 x^7-27}+9}d\sqrt [3]{2 x^7-27}-\frac {1}{18} \log \left (x^7\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{7} \left (\frac {1}{2} \int \frac {1}{x^{14}-3 \sqrt [3]{2 x^7-27}+9}d\sqrt [3]{2 x^7-27}-\frac {1}{18} \log \left (x^7\right )+\frac {1}{6} \log \left (\sqrt [3]{2 x^7-27}+3\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{7} \left (-\int \frac {1}{-x^{14}-27}d\left (2 \sqrt [3]{2 x^7-27}-3\right )-\frac {1}{18} \log \left (x^7\right )+\frac {1}{6} \log \left (\sqrt [3]{2 x^7-27}+3\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{7} \left (\frac {\arctan \left (\frac {2 \sqrt [3]{2 x^7-27}-3}{3 \sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log \left (x^7\right )}{18}+\frac {1}{6} \log \left (\sqrt [3]{2 x^7-27}+3\right )\right )\)

Input:

Int[1/(x*(-27 + 2*x^7)^(2/3)),x]
 

Output:

(ArcTan[(-3 + 2*(-27 + 2*x^7)^(1/3))/(3*Sqrt[3])]/(3*Sqrt[3]) - Log[x^7]/1 
8 + Log[3 + (-27 + 2*x^7)^(1/3)]/6)/7
 

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 70

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

Time = 7.81 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {\ln \left (3+\left (2 x^{7}-27\right )^{\frac {1}{3}}\right )}{63}-\frac {\ln \left (\left (2 x^{7}-27\right )^{\frac {2}{3}}-3 \left (2 x^{7}-27\right )^{\frac {1}{3}}+9\right )}{126}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (2 x^{7}-27\right )^{\frac {1}{3}}}{9}-\frac {\sqrt {3}}{3}\right )}{63}\)	\(67\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {9 \ln \left (3\right )}{2}+7 \ln \left (x \right )+\ln \left (2\right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{7} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], \frac {2 x^{7}}{27}\right )}{81}\right )}{63 \operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}\)	\(74\)
trager	\(\frac {\operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \ln \left (\frac {-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}-48949965800622396478998 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}+24206310434198416909112 x^{7}+347409114848503477844697 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {2}{3}}-56201354332314412587237 \left (2 x^{7}-27\right )^{\frac {2}{3}}-1042227344545510433534091 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {1}{3}}+757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}+168604062996943237761711 \left (2 x^{7}-27\right )^{\frac {1}{3}}+3210832682579892860512479 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-496462116886011762183999}{x^{7}}\right )}{7}-\frac {\ln \left (-\frac {757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}-20295123418338509861200 x^{7}+347409114848503477844697 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {2}{3}}+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}-1042227344545510433534091 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {1}{3}}-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+843871231734515240028696}{x^{7}}\right )}{63}-\frac {\ln \left (-\frac {757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) x^{7}-20295123418338509861200 x^{7}+347409114848503477844697 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {2}{3}}+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}-1042227344545510433534091 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \left (2 x^{7}-27\right )^{\frac {1}{3}}-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+843871231734515240028696}{x^{7}}\right ) \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )}{7}\)	\(452\)

Input:

int(1/x/(2*x^7-27)^(2/3),x,method=_RETURNVERBOSE)
 

Output:

1/63*ln(3+(2*x^7-27)^(1/3))-1/126*ln((2*x^7-27)^(2/3)-3*(2*x^7-27)^(1/3)+9 
)+1/63*3^(1/2)*arctan(2/9*3^(1/2)*(2*x^7-27)^(1/3)-1/3*3^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {2}{9} \, \sqrt {3} {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \] Input:

integrate(1/x/(2*x^7-27)^(2/3),x, algorithm="fricas")
 

Output:

1/63*sqrt(3)*arctan(2/9*sqrt(3)*(2*x^7 - 27)^(1/3) - 1/3*sqrt(3)) - 1/126* 
log((2*x^7 - 27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) + 1/63*log((2*x^7 - 27) 
^(1/3) + 3)
                                                                                    
                                                                                    
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=- \frac {\sqrt [3]{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {27 e^{2 i \pi }}{2 x^{7}}} \right )}}{14 x^{\frac {14}{3}} \Gamma \left (\frac {5}{3}\right )} \] Input:

integrate(1/x/(2*x**7-27)**(2/3),x)
 

Output:

-2**(1/3)*gamma(2/3)*hyper((2/3, 2/3), (5/3,), 27*exp_polar(2*I*pi)/(2*x** 
7))/(14*x**(14/3)*gamma(5/3))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \] Input:

integrate(1/x/(2*x^7-27)^(2/3),x, algorithm="maxima")
 

Output:

1/63*sqrt(3)*arctan(1/9*sqrt(3)*(2*(2*x^7 - 27)^(1/3) - 3)) - 1/126*log((2 
*x^7 - 27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) + 1/63*log((2*x^7 - 27)^(1/3) 
 + 3)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left | {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3 \right |}\right ) \] Input:

integrate(1/x/(2*x^7-27)^(2/3),x, algorithm="giac")
 

Output:

1/63*sqrt(3)*arctan(1/9*sqrt(3)*(2*(2*x^7 - 27)^(1/3) - 3)) - 1/126*log((2 
*x^7 - 27)^(2/3) - 3*(2*x^7 - 27)^(1/3) + 9) + 1/63*log(abs((2*x^7 - 27)^( 
1/3) + 3))
 

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {\ln \left (\frac {{\left (2\,x^7-27\right )}^{1/3}}{49}+\frac {3}{49}\right )}{63}-\ln \left (\frac {27}{14}-\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right )+\ln \left (\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}-\frac {27}{14}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (-\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right ) \] Input:

int(1/(x*(2*x^7 - 27)^(2/3)),x)
 

Output:

log((2*x^7 - 27)^(1/3)/49 + 3/49)/63 - log((3^(1/2)*27i)/14 - (9*(2*x^7 - 
27)^(1/3))/7 + 27/14)*((3^(1/2)*1i)/126 + 1/126) + log((3^(1/2)*27i)/14 + 
(9*(2*x^7 - 27)^(1/3))/7 - 27/14)*((3^(1/2)*1i)/126 - 1/126)
 

Reduce [F]

\[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\int \frac {1}{\left (2 x^{7}-27\right )^{\frac {2}{3}} x}d x \] Input:

int(1/x/(2*x^7-27)^(2/3),x)
 

Output:

int(1/((2*x**7 - 27)**(2/3)*x),x)