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

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

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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, 203} \begin {gather*} \sqrt {2} \tan ^{-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)/((-1 + 3*x + x^2)*Sqrt[-1 + x^3]),x]

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 (-1+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} \tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-1+x^3}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.59, size = 291, normalized size = 9.39 \begin {gather*} \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \sqrt {x^2+x+1} \left (-\frac {\sqrt {3} \left (1+\sqrt [3]{-1}\right ) \left (x+\sqrt [3]{-1}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x-1}+\frac {6 i \left (\left (-1+5 \sqrt [3]{-1}+\sqrt {13}+\sqrt [3]{-1} \sqrt {13}\right ) \Pi \left (\frac {2 \sqrt {3}}{2 i+\sqrt {3}+i \sqrt {13}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )-\left (1-5 \sqrt [3]{-1}+\sqrt {13}+\sqrt [3]{-1} \sqrt {13}\right ) \Pi \left (\frac {2 i \sqrt {3}}{-3+2 \sqrt [3]{-1}+\sqrt {13}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )}{-12-4 i \sqrt {3}}\right )}{3 \sqrt {x^3-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*(-((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[Ar
cSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x)) + ((6*I)*((-1 + 5*(-1)^(1/3
) + Sqrt[13] + (-1)^(1/3)*Sqrt[13])*EllipticPi[(2*Sqrt[3])/(2*I + Sqrt[3] + I*Sqrt[13]), ArcSin[Sqrt[(1 - (-1)
^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)] - (1 - 5*(-1)^(1/3) + Sqrt[13] + (-1)^(1/3)*Sqrt[13])*EllipticPi[((2
*I)*Sqrt[3])/(-3 + 2*(-1)^(1/3) + Sqrt[13]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/
(-12 - (4*I)*Sqrt[3])))/(3*Sqrt[-1 + x^3])

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IntegrateAlgebraic [A]  time = 1.15, size = 31, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3-1}}{x^2+x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - x + 3\right )}}{4 \, \sqrt {x^{3} - 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(x^2 - x + 3)/sqrt(x^3 - 1))

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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} + 3 \, x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 1641, normalized size = 52.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-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))-3/2*13^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))
^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I
*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*
EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/
2-1/2*I*3^(1/2)))^(1/2))-1/2*I*13^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*
3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+
1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/(-
3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/
2))*3^(1/2)+15/2*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2
*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3
/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/
2),(3/2+1/2*I*3^(1/2))/(5/2-1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+5/2*I*(1/(-3/2-1/2*
I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3
^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/
2)/(x^3-1)^(1/2)/(5/2-1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2-1/
2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)+3/2*13^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/
(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2)
)^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2
)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3
/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+1/2*I*13^(1/2)*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))
^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I
*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(1/2))*
EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/
2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)+15/2*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^
(1/2))*x+1/2/(3/2-1/2*I*3^(1/2))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/
2*I*3^(1/2))+1/2*I/(3/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/
2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)
)+5/2*I*(1/(-3/2-1/2*I*3^(1/2))*x-1/(-3/2-1/2*I*3^(1/2)))^(1/2)*(1/(3/2-1/2*I*3^(1/2))*x+1/2/(3/2-1/2*I*3^(1/2
))-1/2*I/(3/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*(1/(3/2+1/2*I*3^(1/2))*x+1/2/(3/2+1/2*I*3^(1/2))+1/2*I/(3/2+1/2*I*
3^(1/2))*3^(1/2))^(1/2)/(x^3-1)^(1/2)/(5/2+1/2*13^(1/2))*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1
/2*I*3^(1/2))/(5/2+1/2*13^(1/2)),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))*3^(1/2)

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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} + 3 \, x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 273, normalized size = 8.81 \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 {\sqrt {13}}{2}+\frac {5}{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 {\sqrt {13}}{2}-\frac {5}{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)*(3*x + x^2 - 1)),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)/(13^(1/2)/
2 + 5/2), 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)) - ell
ipticF(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)) + ellipt
icPi(-((3^(1/2)*1i)/2 + 3/2)/(13^(1/2)/2 - 5/2), 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} + 3 x - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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