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

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [C] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 60, 632, 210, 31} \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {3-2 \sqrt [3]{2 x^7-27}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}+\frac {1}{42} \log \left (\sqrt [3]{2 x^7-27}+3\right )-\frac {\log (x)}{18} \]

[In]

Int[1/(x*(-27 + 2*x^7)^(2/3)),x]

[Out]

-1/21*ArcTan[(3 - 2*(-27 + 2*x^7)^(1/3))/(3*Sqrt[3])]/Sqrt[3] - Log[x]/18 + Log[3 + (-27 + 2*x^7)^(1/3)]/42

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 60

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] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x
)^(1/3)], x] + Dist[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 210

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \text {Subst}\left (\int \frac {1}{x (-27+2 x)^{2/3}} \, dx,x,x^7\right ) \\ & = -\frac {\log (x)}{18}+\frac {1}{42} \text {Subst}\left (\int \frac {1}{3+x} \, dx,x,\sqrt [3]{-27+2 x^7}\right )+\frac {1}{14} \text {Subst}\left (\int \frac {1}{9-3 x+x^2} \, dx,x,\sqrt [3]{-27+2 x^7}\right ) \\ & = -\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\frac {1}{7} \text {Subst}\left (\int \frac {1}{-27-x^2} \, dx,x,-3+2 \sqrt [3]{-27+2 x^7}\right ) \\ & = -\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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{126} \left (-2 \sqrt {3} \arctan \left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )+2 \log \left (3+\sqrt [3]{-27+2 x^7}\right )-\log \left (9-3 \sqrt [3]{-27+2 x^7}+\left (-27+2 x^7\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

(-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

Maple [A] (verified)

Time = 7.37 (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 \(\frac {{\left (-\operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {9 \ln \left (3\right )}{2}+7 \ln \left (x \right )+\ln \left (2\right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{7} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (1,1,\frac {5}{3};2,2;\frac {2 x^{7}}{27}\right )}{81}\right )}{63 \operatorname {signum}\left (-1+\frac {2 x^{7}}{27}\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}\) \(74\)
trager \(-\frac {\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 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+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 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-284407101279777730376610 \left (2 x^{7}-27\right )^{\frac {1}{3}}+843871231734515240028696}{x^{7}}\right ) \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )}{7}+\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 \left (2 x^{7}-27\right )^{\frac {2}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}+1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+56201354332314412587237 \left (2 x^{7}-27\right )^{\frac {2}{3}}-3210832682579892860512479 \operatorname {RootOf}\left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-168604062996943237761711 \left (2 x^{7}-27\right )^{\frac {1}{3}}+496462116886011762183999}{x^{7}}\right )}{7}\) \(453\)

[In]

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

[Out]

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)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {2}{9} \, \sqrt {3} {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \]

[In]

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

[Out]

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 = 0.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=- \frac {\sqrt [3]{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {27 e^{2 i \pi }}{2 x^{7}}} \right )}}{14 x^{\frac {14}{3}} \Gamma \left (\frac {5}{3}\right )} \]

[In]

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

[Out]

-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)

none

Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3\right ) \]

[In]

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

[Out]

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)

none

Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {1}{63} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} - 3\right )}\right ) - \frac {1}{126} \, \log \left ({\left (2 \, x^{7} - 27\right )}^{\frac {2}{3}} - 3 \, {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 9\right ) + \frac {1}{63} \, \log \left ({\left | {\left (2 \, x^{7} - 27\right )}^{\frac {1}{3}} + 3 \right |}\right ) \]

[In]

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

[Out]

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 = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx=\frac {\ln \left (\frac {{\left (2\,x^7-27\right )}^{1/3}}{49}+\frac {3}{49}\right )}{63}-\ln \left (\frac {27}{14}-\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right )+\ln \left (\frac {9\,{\left (2\,x^7-27\right )}^{1/3}}{7}-\frac {27}{14}+\frac {\sqrt {3}\,27{}\mathrm {i}}{14}\right )\,\left (-\frac {1}{126}+\frac {\sqrt {3}\,1{}\mathrm {i}}{126}\right ) \]

[In]

int(1/(x*(2*x^7 - 27)^(2/3)),x)

[Out]

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)