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

Optimal. Leaf size=30 \[ \frac {\sqrt {x^2-1} \sqrt {x^2+1} \sinh ^{-1}(x)}{\sqrt {x^4-1}} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1152, 215} \begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} \sinh ^{-1}(x)}{\sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^2]/Sqrt[-1 + x^4],x]

[Out]

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

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 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]

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 {\sqrt {-1+x^2} \sqrt {1+x^2} \sinh ^{-1}(x)}{\sqrt {-1+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.27 \begin {gather*} \log \left (x^3+\sqrt {x^2-1} \sqrt {x^4-1}-x\right )-\log \left (1-x^2\right ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.69, size = 73, normalized size = 2.43 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} - 1}}{\sqrt {x^{4} - 1}}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x^{4}-1}\, \arcsinh \relax (x )}{\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}} \end {gather*}

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)*arcsinh(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} - 1}}{\sqrt {x^{4} - 1}}\,{d x} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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