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

Optimal. Leaf size=27 \[ \sinh ^{-1}(x)-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+1}}\right ) \]

[Out]

arcsinh(x)-arctanh(x*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {402, 215, 377, 207} \[ \sinh ^{-1}(x)-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/(-1 + x^2),x]

[Out]

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

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 402

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

Rubi steps

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

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Mathematica [B]  time = 0.03, size = 64, normalized size = 2.37 \[ \frac {\log \left (\sqrt {2} \sqrt {x^2+1}-x+1\right )-\log \left (\sqrt {2} \sqrt {x^2+1}+x+1\right )+\log (1-x)-\log (x+1)}{\sqrt {2}}+\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2]/(-1 + x^2),x]

[Out]

ArcSinh[x] + (Log[1 - x] - Log[1 + x] + Log[1 - x + Sqrt[2]*Sqrt[1 + x^2]] - Log[1 + x + Sqrt[2]*Sqrt[1 + x^2]
])/Sqrt[2]

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fricas [B]  time = 0.53, size = 67, normalized size = 2.48 \[ \frac {1}{2} \, \sqrt {2} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {2} {\left (3 \, x^{2} + 1\right )} - 2 \, \sqrt {x^{2} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} + 3}{x^{2} - 1}\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log((9*x^2 - 2*sqrt(2)*(3*x^2 + 1) - 2*sqrt(x^2 + 1)*(3*sqrt(2)*x - 4*x) + 3)/(x^2 - 1)) - log(-x
+ sqrt(x^2 + 1))

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giac [B]  time = 0.62, size = 70, normalized size = 2.59 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 4 \, \sqrt {2} - 6 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 4 \, \sqrt {2} - 6 \right |}}\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 84, normalized size = 3.11 \[ \arcsinh \relax (x )+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2 x +2\right ) \sqrt {2}}{4 \sqrt {-2 x +\left (x +1\right )^{2}}}\right )}{2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {2 x +\left (x -1\right )^{2}}}\right )}{2}-\frac {\sqrt {-2 x +\left (x +1\right )^{2}}}{2}+\frac {\sqrt {2 x +\left (x -1\right )^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((x+1)^2-2*x)^(1/2)+arcsinh(x)+1/2*2^(1/2)*arctanh(1/4*(2-2*x)*2^(1/2)/((x+1)^2-2*x)^(1/2))+1/2*((x-1)^2+
2*x)^(1/2)-1/2*2^(1/2)*arctanh(1/4*(2*x+2)*2^(1/2)/((x-1)^2+2*x)^(1/2))

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maxima [B]  time = 2.98, size = 59, normalized size = 2.19 \[ -\frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x}{{\left | 2 \, x + 2 \right |}} - \frac {2}{{\left | 2 \, x + 2 \right |}}\right ) - \frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x}{{\left | 2 \, x - 2 \right |}} + \frac {2}{{\left | 2 \, x - 2 \right |}}\right ) + \operatorname {arsinh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*arcsinh(2*x/abs(2*x + 2) - 2/abs(2*x + 2)) - 1/2*sqrt(2)*arcsinh(2*x/abs(2*x - 2) + 2/abs(2*x - 2
)) + arcsinh(x)

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mupad [B]  time = 0.17, size = 59, normalized size = 2.19 \[ \mathrm {asinh}\relax (x)+\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{2}-\frac {\sqrt {2}\,\left (\ln \left (x+1\right )-\ln \left (\sqrt {2}\,\sqrt {x^2+1}-x+1\right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)^(1/2)/(x^2 - 1),x)

[Out]

asinh(x) + (2^(1/2)*(log(x - 1) - log(x + 2^(1/2)*(x^2 + 1)^(1/2) + 1)))/2 - (2^(1/2)*(log(x + 1) - log(2^(1/2
)*(x^2 + 1)^(1/2) - x + 1)))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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