3.2.2 \(\int \frac {1+\sqrt {3}-x}{(1-\sqrt {3}-x) \sqrt {1-x^3}} \, dx\) [102]

Optimal. Leaf size=46 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2165, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {2 \sqrt {3}-3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 - x))/Sqrt[1 - x^3]])/Sqrt[-3 + 2*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 2165

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[(1 + k)*(e/d), Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*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^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rubi steps

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

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Mathematica [A]
time = 1.63, size = 49, normalized size = 1.07 \begin {gather*} 2 \sqrt {1+\frac {2}{\sqrt {3}}} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1-x^3}}{1+x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.89, size = 243, normalized size = 5.28

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \ln \left (-\frac {6 \RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \sqrt {3}\, x^{2}+4 \sqrt {3}\, \RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) x +4 \RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right ) \sqrt {3}+48 \sqrt {-x^{3}+1}\, \sqrt {3}+12 \RootOf \left (\textit {\_Z}^{2}-24 \sqrt {3}-36\right )+72 \sqrt {-x^{3}+1}}{\left (x \sqrt {3}+x +2\right )^{2}}\right )}{6}\)	\(134\)
default	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\)	\(243\)
elliptic	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {-1+x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*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))+4*I*(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)/(-3/2+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)/(-3/2+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((1-x+3^(1/2))/(1-x-3^(1/2))/(-x^3+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (35) = 70\).
time = 0.44, size = 207, normalized size = 4.50 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{8} + 16 \, x^{7} + 112 \, x^{6} + 16 \, x^{5} + 112 \, x^{4} - 224 \, x^{3} + 64 \, x^{2} + 4 \, {\left (2 \, x^{6} + 18 \, x^{5} + 42 \, x^{4} + 8 \, x^{3} - \sqrt {3} {\left (x^{6} + 12 \, x^{5} + 18 \, x^{4} + 16 \, x^{3} - 12 \, x^{2} - 8\right )} - 24 \, x + 8\right )} \sqrt {-x^{3} + 1} \sqrt {2 \, \sqrt {3} + 3} - 16 \, \sqrt {3} {\left (x^{7} + 2 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 4 \, x - 4\right )} - 128 \, x + 112}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*sqrt(2*sqrt(3) + 3)*log((x^8 + 16*x^7 + 112*x^6 + 16*x^5 + 112*x^4 - 224*x^3 + 64*x^2 + 4*(2*x^6 +
18*x^5 + 42*x^4 + 8*x^3 - sqrt(3)*(x^6 + 12*x^5 + 18*x^4 + 16*x^3 - 12*x^2 - 8) - 24*x + 8)*sqrt(-x^3 + 1)*sq
rt(2*sqrt(3) + 3) - 16*sqrt(3)*(x^7 + 2*x^6 + 6*x^5 - 5*x^4 + 2*x^3 - 6*x^2 + 4*x - 4) - 128*x + 112)/(x^8 - 8
*x^7 + 16*x^6 + 16*x^5 - 56*x^4 - 32*x^3 + 64*x^2 + 64*x + 16))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - sqrt(3) - 1)/(sqrt(-(x - 1)*(x**2 + x + 1))*(x - 1 + sqrt(3))), 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((1-x+3^(1/2))/(1-x-3^(1/2))/(-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,-3]%%},[2]%%%} / %%%{%%{[-2,4]:[1,0,-3]%%}
,[2]%%%} Er

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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