3.1.46 \(\int \frac {2^{2/3}-2 x}{(2^{2/3}+x) \sqrt {-1-x^3}} \, dx\) [46]

Optimal. Leaf size=39 \[ \frac {2\ 2^{2/3} \tanh ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{\sqrt {3}} \]

[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 = 39, 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 (\sqrt [3]{2} x+1\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 (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 )\\ &=\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 = 1.25, size = 41, normalized size = 1.05 \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 = 1.32, size = 249, normalized size = 6.38

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) \ln \left (\frac {12 \sqrt {-x^{3}-1}\, x -3 \,2^{\frac {2}{3}} x^{2} \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )+\RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) x^{3}+6 \sqrt {-x^{3}-1}\, 2^{\frac {2}{3}}-6 \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x +2\right )^{3}}\right )}{3}\)	\(111\)
default	\(\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(249\)
elliptic	\(\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (2^{\frac {2}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(249\)

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*I*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/
2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(3/
2+1/2*I*3^(1/2)))^(1/2))-2*I*2^(2/3)*3^(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)
))^(1/2)*(-I*(x-1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)/(2^(2/3)+1/2+1/2*I*3^(1/2))*EllipticPi(1/3*3^
(1/2)*(I*(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(2^(2/3)+1/2+1/2*I*3^(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. 241 vs. \(2 (29) = 58\).
time = 0.41, size = 241, normalized size = 6.18 \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 \left (- \frac {2^{\frac {2}{3}}}{x \sqrt {- x^{3} - 1} + 2^{\frac {2}{3}} \sqrt {- x^{3} - 1}}\right )\, 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,[1]%%%} / %%%{%%{[1,0,0]:[1,0,0,-2]%%},[1]%%%} Error:
Bad Argumen

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Mupad [B]
time = 2.84, size = 63, normalized size = 1.62 \begin {gather*} \frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {{\left (\sqrt {3}+\sqrt {-x^3-1}+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(((3^(1/2) + (- x^3 - 1)^(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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