3.30.40 \(\int \frac {(1+x^2) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} (-1+x^2+x^4)} \, dx\) [2940]

3.30.40.1 Optimal result

Integrand size = 42, antiderivative size = 349 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=-\frac {\text {RootSum}\left [1+2 \text {$\#$1}^2-6 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right )-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-3 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}-6 \text {$\#$1}^3-3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{2 \sqrt {2}} \]

output

Unintegrable
 

3.30.40.2 Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.03 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{2 \sqrt {2}} \]

input

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

output

RootSum[1 - 2*#1^2 - 6*#1^4 + 2*#1^6 + #1^8 & , (Log[1 + x^2 + Sqrt[1 + x^ 
4]] - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4 
]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + S 
qrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + Log[1 + x^2 + Sqrt[ 
1 + x^4]]*#1^4 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - S 
qrt[1 + x^4]*#1]*#1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x* 
Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6 
*#1^3 + 3*#1^5 + 2*#1^7) & ]/(2*Sqrt[2])
 

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

Result contains complex when optimal does not.

Time = 2.44 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {7279, 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.

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

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {\sqrt {x^4+1}+x^2}}{\left (2 x^2-\sqrt {5}+1\right ) \sqrt {x^4+1}}+\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {\sqrt {x^4+1}+x^2}}{\left (2 x^2+\sqrt {5}+1\right ) \sqrt {x^4+1}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \left (1+\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {5 \left ((3+i)-(1+i) \sqrt {5}\right )}}+\frac {i \left (1+\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {5 \left ((3+i)-(1+i) \sqrt {5}\right )}}-\frac {\left (1-\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}+\frac {\left (1-\sqrt {5}\right ) \arctan \left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}+i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}\)

input

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

output

((-1/4*I)*(1 + Sqrt[5])*ArcTan[((1 + I)*(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x) 
)/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/Sqrt[5*((3 + I) - (1 
+ I)*Sqrt[5])] + ((I/4)*(1 + Sqrt[5])*ArcTan[((1 + I)*(Sqrt[2] + I*Sqrt[-1 
 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/Sqrt[5* 
((3 + I) - (1 + I)*Sqrt[5])] - ((1 - Sqrt[5])*ArcTan[((1 + I)*(Sqrt[2] - S 
qrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2])])/(4*S 
qrt[(-5 - 5*I)*((2 - I) + Sqrt[5])]) + ((1 - Sqrt[5])*ArcTan[((1 + I)*(Sqr 
t[2] + Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2] 
)])/(4*Sqrt[(-5 - 5*I)*((2 - I) + Sqrt[5])]) - ((1/4 - I/4)*(1 + Sqrt[5])* 
ArcTanh[(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt 
[1 - I*x^2])])/Sqrt[(10 + 10*I)*((-2 - I) + Sqrt[5])] + ((1/4 - I/4)*(1 + 
Sqrt[5])*ArcTanh[(Sqrt[2] + I*Sqrt[-1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt 
[5]]*Sqrt[1 - I*x^2])])/Sqrt[(10 + 10*I)*((-2 - I) + Sqrt[5])] + ((1/4 + I 
/4)*(1 - Sqrt[5])*ArcTanh[(Sqrt[2] - Sqrt[1 + Sqrt[5]]*x)/(Sqrt[(2 + I) + 
I*Sqrt[5]]*Sqrt[1 - I*x^2])])/Sqrt[(10 + 10*I)*((2 + I) + Sqrt[5])] - ((1/ 
4 + I/4)*(1 - Sqrt[5])*ArcTanh[(Sqrt[2] + Sqrt[1 + Sqrt[5]]*x)/(Sqrt[(2 + 
I) + I*Sqrt[5]]*Sqrt[1 - I*x^2])])/Sqrt[(10 + 10*I)*((2 + I) + Sqrt[5])]
 

3.30.40.3.1 Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

3.30.40.4 Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10

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

input

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

output

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

3.30.40.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 16.26 (sec) , antiderivative size = 10165, normalized size of antiderivative = 29.13 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\text {Too large to display} \]

input

integrate((x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1),x, a 
lgorithm="fricas")
 

output

Too large to include
 

3.30.40.6 Sympy [N/A]

Not integrable

Time = 24.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} \left (x^{4} + x^{2} - 1\right )}\, dx \]

input

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

output

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

3.30.40.7 Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

input

integrate((x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1),x, a 
lgorithm="maxima")
 

output

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

3.30.40.8 Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

input

integrate((x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1),x, a 
lgorithm="giac")
 

output

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

3.30.40.9 Mupad [N/A]

Not integrable

Time = 7.50 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^2+1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}\,\left (x^4+x^2-1\right )} \,d x \]

input

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

output

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