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

Optimal. Leaf size=24 \[ -\frac{\sqrt{x^4-1} \sin ^{-1}(x)}{\sqrt{1-x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0105591, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1152, 217, 206} \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right )}{\sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/Sqrt[-1 + x^4],x]

[Out]

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

Rule 1152

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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{\sqrt{1+x^2}}{\sqrt{-1+x^4}} \, dx &=\frac{\left (\sqrt{-1+x^2} \sqrt{1+x^2}\right ) \int \frac{1}{\sqrt{-1+x^2}} \, dx}{\sqrt{-1+x^4}}\\ &=\frac{\left (\sqrt{-1+x^2} \sqrt{1+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-1+x^2}}\right )}{\sqrt{-1+x^4}}\\ &=\frac{\sqrt{-1+x^2} \sqrt{1+x^2} \tanh ^{-1}\left (\frac{x}{\sqrt{-1+x^2}}\right )}{\sqrt{-1+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0201424, size = 34, normalized size = 1.42 \[ \log \left (x^3+\sqrt{x^2+1} \sqrt{x^4-1}+x\right )-\log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 + x^2] + Log[x + x^3 + Sqrt[1 + x^2]*Sqrt[-1 + x^4]]

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Maple [A]  time = 0.01, size = 33, normalized size = 1.4 \begin{align*}{\sqrt{{x}^{4}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{4} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.08377, size = 165, normalized size = 6.88 \begin{align*} \frac{1}{2} \, \log \left (\frac{x^{3} + \sqrt{x^{4} - 1} \sqrt{x^{2} + 1} + x}{x^{3} + x}\right ) - \frac{1}{2} \, \log \left (-\frac{x^{3} - \sqrt{x^{4} - 1} \sqrt{x^{2} + 1} + x}{x^{3} + x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{4} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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