\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx\) [2885]

Optimal result

Integrand size = 32, antiderivative size = 312 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx=\frac {1+x^2+2 x^4+x \left (-1-x^2\right ) \left (x^2+\sqrt {1+x^4}\right )+\sqrt {1+x^4} \left (1+2 x^2-x \left (x^2+\sqrt {1+x^4}\right )\right )}{2 \left (-1+x^2\right ) \left (x^2+\sqrt {1+x^4}\right )^{3/2}}+\frac {\arctan \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {2}}}\right )}{2 \sqrt {-1+\sqrt {2}}}-\frac {1}{2} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+\sqrt {2}}}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{2} \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

1/2*(1+x^2+2*x^4+x*(-x^2-1)*(x^2+(x^4+1)^(1/2))+(x^4+1)^(1/2)*(1+2*x^2-x*(x^2+(x^4+1)^(1/2))))/(x^2-1)/(x^2+(x
^4+1)^(1/2))^(3/2)+1/2*arctan((x^2+(x^4+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)-1/2*(1+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*arctanh((x^2+(x^4+1)^(
1/2))^(1/2)/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)+1/2*(2^(1/2)-1)^(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 = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.40, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2158, 745, 739, 212} \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx=-\frac {1}{4} (1-i)^{3/2} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{4} (1+i)^{3/2} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {\sqrt {1-i x^2}}{2 (x+1)}-\frac {\sqrt {1+i x^2}}{2 (x+1)} \]

[In]

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

[Out]

-1/2*Sqrt[1 - I*x^2]/(1 + x) - Sqrt[1 + I*x^2]/(2*(1 + x)) - ((1 - I)^(3/2)*ArcTanh[(1 + I*x)/(Sqrt[1 - I]*Sqr
t[1 - I*x^2])])/4 - ((1 + I)^(3/2)*ArcTanh[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/4

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 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c*(d/(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

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)^2 \sqrt {1-i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+x)^2 \sqrt {1+i x^2}} \, dx \\ & = -\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}-\frac {1}{2} i \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx+\frac {1}{2} i \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx \\ & = -\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}+\frac {1}{2} i \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} i \text {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-i x}{\sqrt {1+i x^2}}\right ) \\ & = -\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}-\frac {1}{4} (1-i)^{3/2} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{4} (1+i)^{3/2} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx=\frac {1}{2} \left (\frac {-1-2 x^4-\sqrt {1+x^4}-x^2 \left (1+2 \sqrt {1+x^4}\right )}{(1+x) \left (x^2+\sqrt {1+x^4}\right )^{3/2}}+\frac {\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {-1+\sqrt {2}}}-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {\text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {1+\sqrt {2}}}+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )\right ) \]

[In]

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

[Out]

((-1 - 2*x^4 - Sqrt[1 + x^4] - x^2*(1 + 2*Sqrt[1 + x^4]))/((1 + x)*(x^2 + Sqrt[1 + x^4])^(3/2)) + ArcTan[Sqrt[
1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[-1 + Sqrt[2]] - Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(-1 + Sqrt[2])]*
x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]
]/Sqrt[1 + Sqrt[2]] + Sqrt[-1 + Sqrt[2]]*ArcTanh[(Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2
+ Sqrt[1 + x^4])])/2

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (239) = 478\).

Time = 1.23 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx=-\frac {{\left (x + 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (-\frac {\sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2} - 1} + {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - 2 \, \sqrt {x^{4} + 1} \sqrt {-\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) - {\left (x + 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (\frac {\sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2} - 1} - {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - 2 \, \sqrt {x^{4} + 1} \sqrt {-\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) - {\left (x + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) + {\left (x + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - \sqrt {x^{4} + 1} - 1\right )}}{8 \, {\left (x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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