\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(-1+x^4) \sqrt {1+x^4}} \, dx\) [2804]

Optimal result

Integrand size = 34, antiderivative size = 273 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.13, number of steps used = 26, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6857, 2158, 739, 212} \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right ) \]

[In]

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

[Out]

(Sqrt[1 + I]*ArcTanh[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/8 - (Sqrt[1 - I]*ArcTanh[(1 - I*x)/(Sqrt[1 - I]*S
qrt[1 - I*x^2])])/8 + (Sqrt[1 - I]*ArcTanh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/8 - (Sqrt[1 + I]*ArcTanh[
(1 + x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/8 + (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/8 + (
Sqrt[1 + I]*ArcTanh[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/8 - (Sqrt[1 + I]*ArcTanh[(1 + I*x)/(Sqrt[1 + I]*
Sqrt[1 + I*x^2])])/8 - (Sqrt[1 - I]*ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/8

Rule 212

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

Rule 739

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 2158

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 6857

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*} \text {integral}& = \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-x^2\right ) \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+x^2\right ) \sqrt {1+x^4}}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{2} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx \\ & = -\left (\frac {1}{2} \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x) \sqrt {1+x^4}}\right ) \, dx\right )-\frac {1}{2} \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}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i-x) \sqrt {1+x^4}} \, dx\right )-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i+x) \sqrt {1+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1-x) \sqrt {1+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx \\ & = -\left (\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1+i x^2}} \, dx\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1+i x^2}} \, dx-\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(1-x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(1-x) \sqrt {1+i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx \\ & = -\left (\left (-\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-x}{\sqrt {1-i x^2}}\right )\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-x}{\sqrt {1-i x^2}}\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-i x}{\sqrt {1+i x^2}}\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-i x}{\sqrt {1+i x^2}}\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+i x}{\sqrt {1-i x^2}}\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+i x}{\sqrt {1-i x^2}}\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+x}{\sqrt {1+i x^2}}\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \text {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+x}{\sqrt {1+i x^2}}\right ) \\ & = \frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \left (-\arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{4}-1\right ) \sqrt {x^{4}+1}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (201) = 402\).

Time = 9.13 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} - x^{2}\right )} \sqrt {-\sqrt {2} + 1} + 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} - 3 \, x^{4} - 1\right )} \sqrt {-\sqrt {2} + 1}}{x^{4} - 1}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left (\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} - x^{2}\right )} \sqrt {-\sqrt {2} + 1} - 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} - 3 \, x^{4} - 1\right )} \sqrt {-\sqrt {2} + 1}}{x^{4} - 1}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) \]

[In]

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

[Out]

1/16*sqrt(2)*sqrt(-sqrt(2) + 1)*log(-(2*sqrt(x^4 + 1)*(sqrt(2)*x^2 - x^2)*sqrt(-sqrt(2) + 1) + 2*(sqrt(2)*x^3
- 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x - x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 - 3*x^4 - 1)*sqrt(-sqrt(2)
+ 1))/(x^4 - 1)) - 1/16*sqrt(2)*sqrt(-sqrt(2) + 1)*log((2*sqrt(x^4 + 1)*(sqrt(2)*x^2 - x^2)*sqrt(-sqrt(2) + 1
) - 2*(sqrt(2)*x^3 - 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x - x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 - 3*x^4
- 1)*sqrt(-sqrt(2) + 1))/(x^4 - 1)) - 1/16*sqrt(2)*sqrt(sqrt(2) + 1)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 +
1)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) + (2*sqrt(2)*x^4 + 3*x^4 + 2*sqrt(x^4 + 1)*(sqrt(2)*x^2 + x^2)
+ 1)*sqrt(sqrt(2) + 1))/(x^4 - 1)) + 1/16*sqrt(2)*sqrt(sqrt(2) + 1)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1
)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) - (2*sqrt(2)*x^4 + 3*x^4 + 2*sqrt(x^4 + 1)*(sqrt(2)*x^2 + x^2) +
1)*sqrt(sqrt(2) + 1))/(x^4 - 1))

Sympy [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{\left (x^4-1\right )\,\sqrt {x^4+1}} \,d x \]

[In]

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

[Out]

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