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

Optimal. Leaf size=29 \[ -2 \tanh ^{-1}\left (\frac {x}{x^2+\sqrt {x^4-x^2+1}-2 x+1}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 46, normalized size of antiderivative = 1.59, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1741, 12, 1247, 724, 206, 1698} \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right )-\tanh ^{-1}\left (\frac {x^2+1}{2 \sqrt {x^4-x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 1.72, size = 742, normalized size = 25.59 \begin {gather*} \frac {2 \sqrt [3]{-1} \left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )^2 \sqrt {\frac {\sqrt {1-\sqrt [3]{-1}} \left (\sqrt {1+(-1)^{2/3}}-x\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \sqrt {-\frac {\sqrt {1-\sqrt [3]{-1}} \left (x+\sqrt {1+(-1)^{2/3}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) x-\sqrt [3]{-1}}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}} \left (\left (-2+\sqrt [3]{-1}-2 \sqrt {1-\sqrt [3]{-1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (x+\sqrt {1-\sqrt [3]{-1}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}}\right )|\frac {\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )^2}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )^2}\right )+4 \sqrt {1-\sqrt [3]{-1}} \Pi \left (\frac {\left (-1+\sqrt {1-\sqrt [3]{-1}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )}{\left (1+\sqrt {1-\sqrt [3]{-1}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right ) \left (x+\sqrt {1-\sqrt [3]{-1}}\right )}{\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right ) \left (\sqrt {1-\sqrt [3]{-1}}-x\right )}}\right )|\frac {\left (\sqrt {1-\sqrt [3]{-1}}+\sqrt {1+(-1)^{2/3}}\right )^2}{\left (\sqrt {1-\sqrt [3]{-1}}-\sqrt {1+(-1)^{2/3}}\right )^2}\right )\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]

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

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IntegrateAlgebraic [A]  time = 0.37, size = 29, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{1-2 x+x^2+\sqrt {1-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 36, normalized size = 1.24 \begin {gather*} \log \left (\frac {2 \, x^{2} - 3 \, x - \sqrt {x^{4} - x^{2} + 1} + 2}{x^{2} - 2 \, 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]

log((2*x^2 - 3*x - sqrt(x^4 - x^2 + 1) + 2)/(x^2 - 2*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 [A]  time = 0.33, size = 31, normalized size = 1.07

method	result	size
trager	\(\ln \left (-\frac {-2 x^{2}+\sqrt {x^{4}-x^{2}+1}+3 x -2}{\left (-1+x \right )^{2}}\right )\)	\(31\)
elliptic	\(-\arctanh \left (\frac {x^{2}+1}{2 \sqrt {\left (x^{2}-1\right )^{2}+x^{2}}}\right )-\arctanh \left (\frac {\sqrt {x^{4}-x^{2}+1}}{x}\right )\)	\(44\)
default	\(\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2-2 i \sqrt {3}}}{2}\right )}{\sqrt {2+2 i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}-\arctanh \left (\frac {x^{2}+1}{2 \sqrt {x^{4}-x^{2}+1}}\right )-\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}\)	\(210\)

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]

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

[Out]

int((x + 1)/((x - 1)*(x^4 - x^2 + 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}{\left (x - 1\right ) \sqrt {x^{4} - x^{2} + 1}}\, 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)/((x - 1)*sqrt(x**4 - x**2 + 1)), x)

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