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

Optimal. Leaf size=49 \[ -\frac{1}{2} \tan ^{-1}\left (\sqrt{x^2+2 x}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+1}{\sqrt{3} \sqrt{x^2+2 x}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0305797, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {984, 688, 204, 724, 206} \[ -\frac{1}{2} \tan ^{-1}\left (\sqrt{x^2+2 x}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+1}{\sqrt{3} \sqrt{x^2+2 x}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x^2)*Sqrt[2*x + x^2]),x]

[Out]

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

Rule 984

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

Rule 688

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 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])

Rubi steps

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

Mathematica [A]  time = 0.0293119, size = 64, normalized size = 1.31 \[ -\frac{\sqrt{x} \sqrt{x+2} \left (3 \tan ^{-1}\left (\sqrt{\frac{x}{x+2}}\right )+\sqrt{3} \tanh ^{-1}\left (\sqrt{3} \sqrt{\frac{x}{x+2}}\right )\right )}{3 \sqrt{x (x+2)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{2}\arctan \left ({\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-1}}} \right ) }-{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{ \left ( 2+4\,x \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-1+4\,x}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(1/((1+x)^2-1)^(1/2))-1/6*3^(1/2)*arctanh(1/6*(2+4*x)*3^(1/2)/((-1+x)^2-1+4*x)^(1/2))

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Maxima [A]  time = 1.46375, size = 73, normalized size = 1.49 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{2} + 2 \, x}}{{\left | 2 \, x - 2 \right |}} + \frac{6}{{\left | 2 \, x - 2 \right |}} + 2\right ) + \frac{1}{2} \, \arcsin \left (\frac{2}{{\left | 2 \, x + 2 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*log(2*sqrt(3)*sqrt(x^2 + 2*x)/abs(2*x - 2) + 6/abs(2*x - 2) + 2) + 1/2*arcsin(2/abs(2*x + 2))

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Fricas [A]  time = 2.16267, size = 170, normalized size = 3.47 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{\sqrt{3}{\left (2 \, x + 1\right )} + \sqrt{x^{2} + 2 \, x}{\left (2 \, \sqrt{3} - 3\right )} - 4 \, x - 2}{x - 1}\right ) - \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11763, size = 96, normalized size = 1.96 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | -2 \, x - 2 \, \sqrt{3} + 2 \, \sqrt{x^{2} + 2 \, x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{3} + 2 \, \sqrt{x^{2} + 2 \, x} + 2 \right |}}\right ) - \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(abs(-2*x - 2*sqrt(3) + 2*sqrt(x^2 + 2*x) + 2)/abs(-2*x + 2*sqrt(3) + 2*sqrt(x^2 + 2*x) + 2)) -
arctan(-x + sqrt(x^2 + 2*x) - 1)