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

Optimal. Leaf size=73 \[ \frac {\sqrt {x^2-1} \sqrt {x^4-1} \sinh ^{-1}(x)}{\left (1-x^2\right ) \sqrt {x^2+1}}-\frac {\sqrt {x^4-1} \sin ^{-1}(x)}{\sqrt {1-x^2} \sqrt {x^2+1}} \]

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Rubi [A]  time = 0.12, antiderivative size = 72, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 1152, 215, 217, 206} \begin {gather*} \frac {\sqrt {x^2-1} \sqrt {x^2+1} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )}{\sqrt {x^4-1}}-\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^2])/Sqrt[-1 + x^4],x]

[Out]

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

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 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 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 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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {-\sqrt {-1+x^2}+\sqrt {1+x^2}}{\sqrt {-1+x^4}} \, dx &=\int \left (-\frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}}+\frac {\sqrt {1+x^2}}{\sqrt {-1+x^4}}\right ) \, dx\\ &=-\int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx+\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 ) \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}}+\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} \sinh ^{-1}(x)}{\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*}

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Mathematica [A]  time = 0.06, size = 71, normalized size = 0.97 \begin {gather*} \log \left (1-x^2\right )-\log \left (x^2+1\right )-\log \left (x^3+\sqrt {x^2-1} \sqrt {x^4-1}-x\right )+\log \left (x^3+\sqrt {x^2+1} \sqrt {x^4-1}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 4.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-\sqrt {-1+x^2}+\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^2])/Sqrt[-1 + x^4],x]

[Out]

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

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fricas [B]  time = 1.42, size = 137, normalized size = 1.88 \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 ) - \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^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)) - 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^{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^2+1)^(1/2))/(x^4-1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 59, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {x^{4}-1}\, \arcsinh \relax (x )}{\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}}+\frac {\sqrt {x^{4}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{\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^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)+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + 1} - \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^2+1)^(1/2))/(x^4-1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {x^2-1}-\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^2 + 1)^(1/2))/(x^4 - 1)^(1/2),x)

[Out]

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

[Out]

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

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