3.5.20 \(\int \frac {1+x}{(-1+x) \sqrt {1+x^2+x^4}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^4+x^2+1}}{x^2+x+1}\right )}{\sqrt {3}} \]

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Rubi [A]  time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.82, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1741, 12, 1247, 724, 206, 1698} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (x^2+1\right )}{2 \sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/((-1 + x)*Sqrt[1 + x^2 + x^4]),x]

[Out]

-(ArcTanh[(Sqrt[3]*x)/Sqrt[1 + x^2 + x^4]]/Sqrt[3]) - ArcTanh[(Sqrt[3]*(1 + x^2))/(2*Sqrt[1 + x^2 + x^4])]/Sqr
t[3]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 724

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

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 1741

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x,
0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^
2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + b*
x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e
^2 + a*e^4, 0]

Rubi steps

\begin {align*} \int \frac {1+x}{(-1+x) \sqrt {1+x^2+x^4}} \, dx &=\int -\frac {2 x}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {-1-x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx\\ &=-\left (2 \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx\right )-\operatorname {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}-\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}+2 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,-\frac {3 \left (1+x^2\right )}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1+x^2\right )}{2 \sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.41, size = 221, normalized size = 6.50 \begin {gather*} \frac {\sqrt [6]{-1} \sqrt {6} \left (1-i \sqrt {3}\right ) \left (-x+(-1)^{2/3}+1\right )^2 \sqrt {\frac {2 \sqrt {3} x+\sqrt {3}+3 i}{2 i x+\sqrt {3}-i}} \sqrt {\frac {i \left (2 \sqrt {3} x^2+\sqrt {3}+3 i\right )}{\left (\left (\sqrt {3}+i\right ) x-2 i\right )^2}} \left (F\left (\left .\sin ^{-1}\left (\sqrt {\frac {-2 i x+\sqrt {3}+i}{4 i-2 \left (i+\sqrt {3}\right ) x}}\right )\right |4\right )-2 \Pi \left (-2;\left .\sin ^{-1}\left (\sqrt {\frac {-2 i x+\sqrt {3}+i}{4 i-2 \left (i+\sqrt {3}\right ) x}}\right )\right |4\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x)/((-1 + x)*Sqrt[1 + x^2 + x^4]),x]

[Out]

((-1)^(1/6)*Sqrt[6]*(1 - I*Sqrt[3])*(1 + (-1)^(2/3) - x)^2*Sqrt[(3*I + Sqrt[3] + 2*Sqrt[3]*x)/(-I + Sqrt[3] +
(2*I)*x)]*Sqrt[(I*(3*I + Sqrt[3] + 2*Sqrt[3]*x^2))/(-2*I + (I + Sqrt[3])*x)^2]*(EllipticF[ArcSin[Sqrt[(I + Sqr
t[3] - (2*I)*x)/(4*I - 2*(I + Sqrt[3])*x)]], 4] - 2*EllipticPi[-2, ArcSin[Sqrt[(I + Sqrt[3] - (2*I)*x)/(4*I -
2*(I + Sqrt[3])*x)]], 4]))/((1 + (-1)^(1/3))*Sqrt[1 + x^2 + x^4])

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IntegrateAlgebraic [A]  time = 0.44, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {1+x^2+x^4}}{1+x+x^2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[1 + x^2 + x^4]),x]

[Out]

(-2*ArcTanh[(Sqrt[3]*Sqrt[1 + x^2 + x^4])/(1 + x + x^2)])/Sqrt[3]

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fricas [B]  time = 0.51, size = 73, normalized size = 2.15 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\frac {7 \, x^{4} - 4 \, x^{3} - 2 \, \sqrt {3} \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - x + 2\right )} + 12 \, x^{2} - 4 \, x + 7}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(-1+x)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-(7*x^4 - 4*x^3 - 2*sqrt(3)*sqrt(x^4 + x^2 + 1)*(2*x^2 - x + 2) + 12*x^2 - 4*x + 7)/(x^4 - 4*x
^3 + 6*x^2 - 4*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^{4} + x^{2} + 1} {\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^4+x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x + 1)/(sqrt(x^4 + x^2 + 1)*(x - 1)), x)

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maple [C]  time = 0.30, size = 58, normalized size = 1.71

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {x^{4}+x^{2}+1}}{\left (-1+x \right )^{2}}\right )}{3}\)	\(58\)
elliptic	\(-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (3 x^{2}+3\right ) \sqrt {3}}{6 \sqrt {\left (x^{2}-1\right )^{2}+3 x^{2}}}\right )}{3}-\frac {\sqrt {6}\, \arctanh \left (\frac {\sqrt {6}\, \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{6 x}\right ) \sqrt {2}}{6}\)	\(65\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (3 x^{2}+3\right ) \sqrt {3}}{6 \sqrt {x^{4}+x^{2}+1}}\right )}{3}-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(-1+x)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} + x^{2} + 1} {\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^4+x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/(sqrt(x^4 + x^2 + 1)*(x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x+1}{\left (x-1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/((x - 1)*(x^2 + x^4 + 1)^(1/2)),x)

[Out]

int((x + 1)/((x - 1)*(x^2 + x^4 + 1)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {\left (x^{2} - x + 1\right ) \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**4+x**2+1)**(1/2),x)

[Out]

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

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