\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx\) [2602]

Optimal result

Integrand size = 32, antiderivative size = 225 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

1/2*(-2+2*2^(1/2))^(1/2)*arctan((x^2+(x^4+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))-1/2*(-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)))-1/2*(2+2*2^(1/2))^(1/2)*arctanh((x^2+(x
^4+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))+1/2*(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 = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.36, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2158, 739, 212} \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=-\frac {1}{2} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \]

[In]

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

[Out]

-1/2*(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])])/2

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]

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

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \left (\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (167) = 334\).

Time = 3.32 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.24 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 2} \log \left (-\frac {\sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 2} + {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {-2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 2} \log \left (\frac {\sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 2} - {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {-2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) \]

[In]

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

[Out]

1/8*sqrt(-2*sqrt(2) + 2)*log(-(sqrt(x^4 + 1)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 2) + (2*x^3 + sqrt(2)*(x^3 - x^2
- x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (x^2 + sqrt(2)*(x^2 + 1) + 1
)*sqrt(-2*sqrt(2) + 2))/(x^2 + 2*x + 1)) - 1/8*sqrt(-2*sqrt(2) + 2)*log((sqrt(x^4 + 1)*(sqrt(2) + 2)*sqrt(-2*s
qrt(2) + 2) - (2*x^3 + sqrt(2)*(x^3 - x^2 - x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x) - 2)*sqrt(x^2 + sqr
t(x^4 + 1)) - (x^2 + sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*sqrt(2) + 2))/(x^2 + 2*x + 1)) - 1/8*sqrt(2*sqrt(2) + 2)*l
og(-((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1
)) + (x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1)) + 1/8*s
qrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqr
t(x^2 + sqrt(x^4 + 1)) - (x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2
+ 2*x + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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