Integrand size = 45, antiderivative size = 394 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {\text {RootSum}\left [1+2 \text {$\#$1}^2-6 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right )+\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}-6 \text {$\#$1}^3-3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{\sqrt {2}} \] Output:
Unintegrable
Time = 0.62 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.01 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{\sqrt {2}} \] Input:
Integrate[((1 + x^2 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]
Output:
(2*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] )] + 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] + Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[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 - Sqrt[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) & ])/Sqrt[2]
Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, 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.
\(\displaystyle \int \frac {\left (x^4+x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1} \left (x^4+x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}+\frac {2 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1} \left (x^4+x^2-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \arctan \left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{\sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}+\frac {i \arctan \left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{\sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}-\frac {\arctan \left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{\sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}+\frac {\arctan \left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{\sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}-\frac {(1-i) \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {(1-i) \text {arctanh}\left (\frac {\sqrt {2}+i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {(1+i) \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}-\frac {(1+i) \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}}\) |
Input:
Int[((1 + x^2 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]
Output:
((-I)*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 - 5*I)*((-2 + I) + Sqrt[5])] + (I*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 - 5*I)*((-2 + I) + Sqrt[5])] - A rcTan[((1 + I)*(Sqrt[2] - Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[ 5]]*Sqrt[1 + I*x^2])]/Sqrt[(-5 - 5*I)*((2 - I) + Sqrt[5])] + ArcTan[((1 + I)*(Sqrt[2] + Sqrt[1 + Sqrt[5]]*x))/(Sqrt[(-2 - 4*I) - 2*Sqrt[5]]*Sqrt[1 + I*x^2])]/Sqrt[(-5 - 5*I)*((2 - I) + Sqrt[5])] - ((1 - I)*ArcTanh[(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x)/(Sqrt[(2 + I) - I*Sqrt[5]]*Sqrt[1 - I*x^2])])/S qrt[(10 + 10*I)*((-2 - I) + Sqrt[5])] + ((1 - I)*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 + I)*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 + I)*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] )] + ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]
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]
Not integrable
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10
\[\int \frac {\left (x^{4}+x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )}d x\]
Input:
int((x^4+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^4+x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1),x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 12.97 (sec) , antiderivative size = 9805, normalized size of antiderivative = 24.89 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate((x^4+x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1), x, algorithm="fricas")
Output:
Too large to include
Not integrable
Time = 36.95 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.12 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt {x^{4} + 1} \left (x^{4} + x^{2} - 1\right )}\, dx \] Input:
integrate((x**4+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(sqrt(x**2 + sqrt(x**4 + 1))*(x**2 - x + 1)*(x**2 + x + 1)/(sqrt(x **4 + 1)*(x**4 + x**2 - 1)), x)
Not integrable
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.10 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^4+x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1), x, algorithm="maxima")
Output:
integrate((x^4 + x^2 + 1)*sqrt(x^2 + sqrt(x^4 + 1))/((x^4 + x^2 - 1)*sqrt( x^4 + 1)), x)
Not integrable
Time = 0.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.10 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^4+x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1), x, algorithm="giac")
Output:
integrate((x^4 + x^2 + 1)*sqrt(x^2 + sqrt(x^4 + 1))/((x^4 + x^2 - 1)*sqrt( x^4 + 1)), x)
Not integrable
Time = 9.81 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.10 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}\,\left (x^4+x^2+1\right )}{\sqrt {x^4+1}\,\left (x^4+x^2-1\right )} \,d x \] Input:
int((((x^4 + 1)^(1/2) + x^2)^(1/2)*(x^2 + x^4 + 1))/((x^4 + 1)^(1/2)*(x^2 + x^4 - 1)),x)
Output:
int((((x^4 + 1)^(1/2) + x^2)^(1/2)*(x^2 + x^4 + 1))/((x^4 + 1)^(1/2)*(x^2 + x^4 - 1)), x)
Not integrable
Time = 200.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.10 \[ \int \frac {\left (1+x^2+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^{4}+x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )}d x \] Input:
int((x^4+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^4+x^2+1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1),x)