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

Optimal. Leaf size=24 \[ -2 \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 46, normalized size of antiderivative = 1.92, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2056, 6733, 1698, 203} \begin {gather*} -\frac {2 \sqrt {x} \sqrt {x^2+x+1} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{\sqrt {x^3+x^2+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x]*Sqrt[1 + x + x^2]*ArcTan[Sqrt[x]/Sqrt[1 + x + x^2]])/Sqrt[x + x^2 + 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 1698

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.30, size = 107, normalized size = 4.46 \begin {gather*} \frac {2 (-1)^{2/3} \sqrt {\frac {\sqrt [3]{-1}}{x}+1} \sqrt {1-\frac {(-1)^{2/3}}{x}} x^{3/2} \left (F\left (i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )-2 \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )\right )}{\sqrt {x \left (x^2+x+1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)/x]*Sqrt[1 - (-1)^(2/3)/x]*x^(3/2)*(EllipticF[I*ArcSinh[(-1)^(5/6)/Sqrt[x]],
(-1)^(2/3)] - 2*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)]))/Sqrt[x*(1 + x + x^2)]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 24, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 18, normalized size = 0.75 \begin {gather*} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.34, size = 45, normalized size = 1.88

method	result	size
trager	\(-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}+x^{2}+x}}{\left (1+x \right )^{2}}\right )\)	\(45\)
default	\(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}-\frac {4 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(271\)
elliptic	\(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}-\frac {4 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.30, size = 179, normalized size = 7.46 \begin {gather*} \frac {\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-2\,\Pi \left (\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,1{}\mathrm {i}}{\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________