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

Optimal. Leaf size=132 \[ \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [C]  time = 0.72, antiderivative size = 161, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6725, 2133, 725, 206} \begin {gather*} -\frac {1}{4} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{4} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{4} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{4} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(Sqrt[1 - I]*ArcTanh[(1 - I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])]) + (Sqrt[1 - I]*ArcTanh[(1 + I*x)/(Sqrt[1 -
I]*Sqrt[1 - I*x^2])])/4 + (Sqrt[1 + I]*ArcTanh[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/4 - (Sqrt[1 + I]*Arc
Tanh[(1 + I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/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 725

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

Rule 2133

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_S
ymbol] :> Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sq
rt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.58, size = 190, normalized size = 1.44 \begin {gather*} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 5.61, size = 289, normalized size = 2.19 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} + x^{2} - \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 3} + \sqrt {2} + 1\right )} + {\left (x^{2} + \sqrt {2} {\left (x^{2} - 2\right )} - 3\right )} \sqrt {-2 \, \sqrt {2} + 3} - 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, x}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} - \sqrt {2} x - \sqrt {x^{4} + 1} x - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} - \sqrt {2} x - \sqrt {x^{4} + 1} x - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*sqrt(sqrt(2) - 1)*arctan(-1/2*(sqrt(2)*x^2 + x^2 - sqrt(x^4 + 1)*((sqrt(2) + 1)*sqrt(-2*sqrt(2) +
3) + sqrt(2) + 1) + (x^2 + sqrt(2)*(x^2 - 2) - 3)*sqrt(-2*sqrt(2) + 3) - 1)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sq
rt(2) - 1)/x) + 1/8*sqrt(2)*sqrt(sqrt(2) + 1)*log(-(sqrt(2)*x^2 + 2*x^2 + (x^3 - sqrt(2)*x - sqrt(x^4 + 1)*x -
x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) + 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 - 1)) - 1/8*sqrt(2)*sqrt(s
qrt(2) + 1)*log(-(sqrt(2)*x^2 + 2*x^2 - (x^3 - sqrt(2)*x - sqrt(x^4 + 1)*x - x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt
(sqrt(2) + 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{2}-1\right ) \sqrt {x^{4}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^4 + 1)^(1/2) + x^2)^(1/2)/((x^2 - 1)*(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} + \sqrt {x^{4} + 1}}}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 + sqrt(x**4 + 1))/((x - 1)*(x + 1)*sqrt(x**4 + 1)), x)

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