3.1.45 \(\int \frac {2^{2/3}+2 x}{(2^{2/3}-x) \sqrt {-1+x^3}} \, dx\) [45]

Optimal. Leaf size=38 \[ -\frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{\sqrt {3}} \]

[Out]

-2/3*2^(2/3)*arctanh((1-2^(1/3)*x)*3^(1/2)/(x^3-1)^(1/2))*3^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2162, 212} \begin {gather*} -\frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {x^3-1}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/Sqrt[3]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2162

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[2*(e/d), Subst[Int[
1/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c))/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rubi steps

\begin {align*} \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx &=-\left (\left (2\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {1-\sqrt [3]{2} x}{\sqrt {-1+x^3}}\right )\right )\\ &=-\frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 1.26, size = 39, normalized size = 1.03 \begin {gather*} \frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {-1+x^3}}{\sqrt {3} \left (-1+\sqrt [3]{2} x\right )}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*2^(2/3)*ArcTanh[Sqrt[-1 + x^3]/(Sqrt[3]*(-1 + 2^(1/3)*x))])/Sqrt[3]

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.36, size = 262, normalized size = 6.89

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) \ln \left (-\frac {3 \,2^{\frac {2}{3}} x^{2} \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )-12 \sqrt {x^{3}-1}\, x +\RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) x^{3}-6 \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x +6 \sqrt {x^{3}-1}\, 2^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x -2\right )^{3}}\right )}{3}\)	\(108\)
default	\(-\frac {4 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {6 \,2^{\frac {2}{3}} \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(262\)
elliptic	\(-\frac {4 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {6 \,2^{\frac {2}{3}} \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(262\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*
((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),
((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-6*2^(2/3)*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^
(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3
-1)^(1/2)/(-2^(2/3)+1)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(-2^(2/3)+1),((3/2+1
/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((2*x + 2^(2/3))/(sqrt(x^3 - 1)*(x - 2^(2/3))), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (28) = 56\).
time = 0.43, size = 238, normalized size = 6.26 \begin {gather*} \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \log \left (\frac {x^{18} + 1440 \, x^{15} + 17400 \, x^{12} - 21056 \, x^{9} - 10368 \, x^{6} + 15360 \, x^{3} + 2 \, \sqrt {6} 2^{\frac {1}{6}} {\left (126 \, x^{14} + 2664 \, x^{11} - 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac {2}{3}} {\left (x^{16} + 310 \, x^{13} + 2332 \, x^{10} - 2656 \, x^{7} - 256 \, x^{4} + 512 \, x\right )} + 2^{\frac {1}{3}} {\left (17 \, x^{15} + 1058 \, x^{12} + 2528 \, x^{9} - 5408 \, x^{6} + 2560 \, x^{3} - 512\right )}\right )} \sqrt {x^{3} - 1} + 24 \cdot 2^{\frac {2}{3}} {\left (x^{17} + 121 \, x^{14} + 478 \, x^{11} - 1144 \, x^{8} + 608 \, x^{5} - 64 \, x^{2}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} + 176 \, x^{13} + 83 \, x^{10} - 680 \, x^{7} + 544 \, x^{4} - 128 \, x\right )} - 2048}{x^{18} - 24 \, x^{15} + 240 \, x^{12} - 1280 \, x^{9} + 3840 \, x^{6} - 6144 \, x^{3} + 4096}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(6)*2^(1/6)*log((x^18 + 1440*x^15 + 17400*x^12 - 21056*x^9 - 10368*x^6 + 15360*x^3 + 2*sqrt(6)*2^(1/6)
*(126*x^14 + 2664*x^11 - 4608*x^5 + 2304*x^2 + 2^(2/3)*(x^16 + 310*x^13 + 2332*x^10 - 2656*x^7 - 256*x^4 + 512
*x) + 2^(1/3)*(17*x^15 + 1058*x^12 + 2528*x^9 - 5408*x^6 + 2560*x^3 - 512))*sqrt(x^3 - 1) + 24*2^(2/3)*(x^17 +
121*x^14 + 478*x^11 - 1144*x^8 + 608*x^5 - 64*x^2) + 48*2^(1/3)*(5*x^16 + 176*x^13 + 83*x^10 - 680*x^7 + 544*
x^4 - 128*x) - 2048)/(x^18 - 24*x^15 + 240*x^12 - 1280*x^9 + 3840*x^6 - 6144*x^3 + 4096))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {2^{\frac {2}{3}}}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx - \int \frac {2 x}{x \sqrt {x^{3} - 1} - 2^{\frac {2}{3}} \sqrt {x^{3} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(2**(2/3)/(x*sqrt(x**3 - 1) - 2**(2/3)*sqrt(x**3 - 1)), x) - Integral(2*x/(x*sqrt(x**3 - 1) - 2**(2/3
)*sqrt(x**3 - 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[2]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Ba
d Argument

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Mupad [B]
time = 2.86, size = 62, normalized size = 1.63 \begin {gather*} \frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {{\left (\sqrt {x^3-1}-\sqrt {3}+2^{1/3}\,\sqrt {3}\,x\right )}^3\,\left (\sqrt {3}+\sqrt {x^3-1}-2^{1/3}\,\sqrt {3}\,x\right )}{{\left (x-2^{2/3}\right )}^6}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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