3.24.75 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x^2)^2 \sqrt {1+x^4}} \, dx\) [2375]

3.24.75.1 Optimal result

Integrand size = 34, antiderivative size = 189 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=\frac {-x^2 \left (-1+x^2\right )-x^2 \sqrt {1+x^4}}{4 x \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{4} \sqrt {-1+5 \sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {1}{4} \sqrt {1+5 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

output

1/4*(-x^2*(x^2-1)-(x^4+1)^(1/2)*x^2)/x/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2)+1 
/4*(-1+5*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*(1+5*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)))
 

3.24.75.2 Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=\frac {1}{4} \left (-\frac {x \left (-1+x^2+\sqrt {1+x^4}\right )}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {-1+5 \sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {1+5 \sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]

input

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

output

(-((x*(-1 + x^2 + Sqrt[1 + x^4]))/((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])) + 
 Sqrt[-1 + 5*Sqrt[2]]*ArcTan[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x 
^4]))/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[1 + 5*Sqrt[2]]*ArcTanh[(-1 + x 
^2 + Sqrt[1 + x^4])/(Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])])/ 
4
 

3.24.75.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{2 \left (-x^2-1\right ) \sqrt {x^4+1}}-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{4 (-x+i)^2 \sqrt {x^4+1}}-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{4 (x+i)^2 \sqrt {x^4+1}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{4 (1+i)^{5/2}}+\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {\text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{4 (1+i)^{5/2}}-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{4 (1-i)^{5/2}}+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {\text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{4 (1-i)^{5/2}}+\frac {i \sqrt {1-i x^2}}{8 (-x+i)}-\frac {i \sqrt {1-i x^2}}{8 (x+i)}-\frac {i \sqrt {1+i x^2}}{8 (-x+i)}+\frac {i \sqrt {1+i x^2}}{8 (x+i)}\)

input

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

output

((I/8)*Sqrt[1 - I*x^2])/(I - x) - ((I/8)*Sqrt[1 - I*x^2])/(I + x) - ((I/8) 
*Sqrt[1 + I*x^2])/(I - x) + ((I/8)*Sqrt[1 + I*x^2])/(I + x) + ArcTanh[(1 - 
 x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])]/(4*(1 + I)^(5/2)) - (Sqrt[1 + I]*ArcTan 
h[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/8 - ArcTanh[(1 + x)/(Sqrt[1 + I] 
*Sqrt[1 - I*x^2])]/(4*(1 + I)^(5/2)) + (Sqrt[1 + I]*ArcTanh[(1 + x)/(Sqrt[ 
1 + I]*Sqrt[1 - I*x^2])])/8 + ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2] 
)]/(4*(1 - I)^(5/2)) - (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + 
I*x^2])])/8 - ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])]/(4*(1 - I)^(5 
/2)) + (Sqrt[1 - I]*ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/8
 

3.24.75.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

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

3.24.75.4 Maple [F]

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

input

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

output

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

3.24.75.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (147) = 294\).

Time = 4.09 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=\frac {{\left (x^{2} + 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{2} + 14 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} - \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x + x\right )} + 5 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 1} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{2} + 14 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} - \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x + x\right )} + 5 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 1} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {-5 \, \sqrt {2} + 1} \log \left (-\frac {7 \, \sqrt {2} x^{2} - 14 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x - x\right )} \sqrt {-5 \, \sqrt {2} + 1} + {\left (x^{3} - \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} + 5 \, x\right )} \sqrt {-5 \, \sqrt {2} + 1}\right )} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + {\left (x^{2} + 1\right )} \sqrt {-5 \, \sqrt {2} + 1} \log \left (-\frac {7 \, \sqrt {2} x^{2} - 14 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x - x\right )} \sqrt {-5 \, \sqrt {2} + 1} + {\left (x^{3} - \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} + 5 \, x\right )} \sqrt {-5 \, \sqrt {2} + 1}\right )} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) - 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, {\left (x^{2} + 1\right )}} \]

input

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

output

1/16*((x^2 + 1)*sqrt(5*sqrt(2) + 1)*log((7*sqrt(2)*x^2 + 14*x^2 + (x^3 + s 
qrt(2)*(2*x^3 + 3*x) - sqrt(x^4 + 1)*(2*sqrt(2)*x + x) + 5*x)*sqrt(x^2 + s 
qrt(x^4 + 1))*sqrt(5*sqrt(2) + 1) + 7*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 
1)) - (x^2 + 1)*sqrt(5*sqrt(2) + 1)*log((7*sqrt(2)*x^2 + 14*x^2 - (x^3 + s 
qrt(2)*(2*x^3 + 3*x) - sqrt(x^4 + 1)*(2*sqrt(2)*x + x) + 5*x)*sqrt(x^2 + s 
qrt(x^4 + 1))*sqrt(5*sqrt(2) + 1) + 7*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 
1)) - (x^2 + 1)*sqrt(-5*sqrt(2) + 1)*log(-(7*sqrt(2)*x^2 - 14*x^2 + sqrt(x 
^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(2*sqrt(2)*x - x)*sqrt(-5*sqrt(2) + 1) 
+ (x^3 - sqrt(2)*(2*x^3 + 3*x) + 5*x)*sqrt(-5*sqrt(2) + 1)) + 7*sqrt(x^4 + 
 1)*(sqrt(2) - 1))/(x^2 + 1)) + (x^2 + 1)*sqrt(-5*sqrt(2) + 1)*log(-(7*sqr 
t(2)*x^2 - 14*x^2 - sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(2*sqrt(2)*x 
- x)*sqrt(-5*sqrt(2) + 1) + (x^3 - sqrt(2)*(2*x^3 + 3*x) + 5*x)*sqrt(-5*sq 
rt(2) + 1)) + 7*sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 + 1)) - 4*(x^3 - sqrt(x^ 
4 + 1)*x + x)*sqrt(x^2 + sqrt(x^4 + 1)))/(x^2 + 1)
 

3.24.75.6 Sympy [F]

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

input

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

output

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

3.24.75.7 Maxima [F]

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

input

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

output

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

3.24.75.8 Giac [F]

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

input

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

output

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

3.24.75.9 Mupad [F(-1)]

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

input

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

output

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