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

Optimal. Leaf size=59 \[ -\frac {\tan ^{-1}\left (\frac {3-2 \sqrt [3]{-27+2 x^7}}{3 \sqrt {3}}\right )}{21 \sqrt {3}}-\frac {\log (x)}{18}+\frac {1}{42} \log \left (3+\sqrt [3]{-27+2 x^7}\right ) \]

[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]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 60, 632, 210, 31} \begin {gather*} -\frac {\text {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{x \left (-27+2 x^7\right )^{2/3}} \, dx &=\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 {\tan ^{-1}\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]
time = 0.06, size = 82, normalized size = 1.39 \begin {gather*} \frac {1}{126} \left (-2 \sqrt {3} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 6.23, size = 74, normalized size = 1.25

method	result	size
meijerg	\(\frac {\left (-\mathrm {signum}\left (-1+\frac {2 x^{7}}{27}\right )\right )^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {9 \ln \left (3\right )}{2}+7 \ln \left (x \right )+\ln \left (2\right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{7} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], \frac {2 x^{7}}{27}\right )}{81}\right )}{63 \mathrm {signum}\left (-1+\frac {2 x^{7}}{27}\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}\)	\(74\)
trager	\(\frac {\RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right ) \ln \left (-\frac {757355840490254039191854 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+48949965800622396478998 \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}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}+1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+56201354332314412587237 \left (2 x^{7}-27\right )^{\frac {2}{3}}-3210832682579892860512479 \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}-\frac {\ln \left (-\frac {757355840490254039191854 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \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}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \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 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2} x^{7}+119351332086100723341414 \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}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )-757355840490254039191854 \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )^{2}-1042227344545510433534091 \left (2 x^{7}-27\right )^{\frac {1}{3}} \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )+94802367093259243458870 \left (2 x^{7}-27\right )^{\frac {2}{3}}+3042531384693169740692067 \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 ) \RootOf \left (81 \textit {\_Z}^{2}+9 \textit {\_Z} +1\right )}{7}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/63/signum(-1+2/27*x^7)^(2/3)*(-signum(-1+2/27*x^7))^(2/3)*((1/6*Pi*3^(1/2)-9/2*ln(3)+7*ln(x)+ln(2)+I*Pi)*GAM
MA(2/3)+4/81*GAMMA(2/3)*x^7*hypergeom([1,1,5/3],[2,2],2/27*x^7))/GAMMA(2/3)

________________________________________________________________________________________

Maxima [A]
time = 2.31, size = 64, normalized size = 1.08 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]
time = 0.98, size = 66, normalized size = 1.12 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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] Result contains complex when optimal does not.
time = 0.45, size = 42, normalized size = 0.71 \begin {gather*} - \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]
time = 0.80, size = 65, normalized size = 1.10 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.46, size = 76, normalized size = 1.29 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________