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

Optimal. Leaf size=21 \[ -\frac {2}{3} \tan ^{-1}\left (\frac {3 \sqrt {x^3-1}}{(x-1)^2}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2138, 203} \begin {gather*} \frac {2}{3} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/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 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rubi steps

\begin {align*} \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{9+x^2} \, dx,x,\frac {(1-x)^2}{\sqrt {-1+x^3}}\right )\\ &=\frac {2}{3} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.19 \begin {gather*} \frac {2}{3} \tan ^{-1}\left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.82, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \tan ^{-1}\left (\frac {3 \sqrt {-1+x^3}}{(-1+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 40, normalized size = 1.90 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt {x^{3} - 1}}{6 \, {\left (x^{4} - x^{3} - x + 1\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)*sqrt(x^3 - 1)/(x^4 - x^3 - x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.35, size = 74, normalized size = 3.52

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-12 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -6 \sqrt {x^{3}-1}\, x -10 \RootOf \left (\textit {\_Z}^{2}+1\right )+6 \sqrt {x^{3}-1}}{\left (2+x \right )^{3}}\right )}{3}\)	\(74\)
default	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(240\)
elliptic	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^3-12*RootOf(_Z^2+1)*x^2-6*RootOf(_Z^2+1)*x-6*(x^3-1)^(1/2)*x-10*RootOf
(_Z^2+1)+6*(x^3-1)^(1/2))/(2+x)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.30, size = 206, normalized size = 9.81 \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}}}\,\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 {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\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 {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\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((x - 1)/((x^3 - 1)^(1/2)*(x + 2)),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)*(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)) - ellipticPi((3^(1/2)*1i)/6 + 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)))*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2))/(((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 - 1}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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