3.4.73 \(\int \frac {-2-2 x+x^2}{(3-x+x^2) \sqrt {-1+x^3}} \, dx\)

Optimal. Leaf size=31 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3-1}}{x^2+x+1}\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2145, 206} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {x^3-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 2145

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + (2*h*x)/g)/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]

Rubi steps

\begin {align*} \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx &=4 \operatorname {Subst}\left (\int \frac {1}{2-4 x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-1+x^3}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.72, size = 496, normalized size = 16.00 \begin {gather*} \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {(-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-3+2 i \sqrt {3}+\sqrt {33}}-\frac {i \sqrt {11} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}-\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{2 i-\sqrt {3}+\sqrt {11}}-\frac {(-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{3-2 i \sqrt {3}+\sqrt {33}}-\frac {i \sqrt {11} \sqrt {x^2+x+1} \Pi \left (\frac {2 \sqrt {3}}{-2 i+\sqrt {3}+\sqrt {11}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-2 i+\sqrt {3}+\sqrt {11}}+\frac {\left (x+\sqrt [3]{-1}\right ) \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{\sqrt {x^3-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*Ellipt
icF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]
- (I*Sqrt[11]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[3] - Sqrt[11]), ArcSin[Sqrt[(1 - (-1)^(2/3
)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(2*I - Sqrt[3] + Sqrt[11]) + ((-1)^(5/6)*(1 + (-1)^(1/3))*Sqrt[1 + x + x
^2]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[3] - Sqrt[11]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1
)^(1/3)])/(-3 + (2*I)*Sqrt[3] + Sqrt[33]) - (I*Sqrt[11]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[
3] + Sqrt[11]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2*I + Sqrt[3] + Sqrt[11]) -
((-1)^(5/6)*(1 + (-1)^(1/3))*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-2*I + Sqrt[3] + Sqrt[11]), ArcSin[Sqrt
[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(3 - (2*I)*Sqrt[3] + Sqrt[33])))/Sqrt[-1 + x^3]

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IntegrateAlgebraic [A]  time = 1.05, size = 31, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 65, normalized size = 2.10 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 14 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} - 1} {\left (x^{2} + 3 \, x - 1\right )} + 7 \, x^{2} - 6 \, x - 7}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log((x^4 + 14*x^3 - 4*sqrt(2)*sqrt(x^3 - 1)*(x^2 + 3*x - 1) + 7*x^2 - 6*x - 7)/(x^4 - 2*x^3 + 7*x^
2 - 6*x + 9))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.52, size = 58, normalized size = 1.87

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{3}-1}}{x^{2}-x +3}\right )}{2}\)	\(58\)
default	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\)	\(423\)
elliptic	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\)	\(423\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*RootOf(_Z^2-2)*ln((RootOf(_Z^2-2)*x^2+3*RootOf(_Z^2-2)*x-RootOf(_Z^2-2)+4*(x^3-1)^(1/2))/(x^2-x+3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 275, normalized size = 8.87 \begin {gather*} \frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3^(1/2)*1i + 3)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(
1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((11^(1/2)
*1i)/2 + 1/2), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))
- ellipticF(asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) + e
llipticPi(-((3^(1/2)*1i)/2 + 3/2)/((11^(1/2)*1i)/2 - 1/2), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3
^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i
)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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