Integrand size = 32, antiderivative size = 119 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] Output:
1/2*x*(x^2+(x^4+1)^(1/2))^(1/2)-1/2*arctan(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^( 1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)-arctanh(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^ (1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {1}{2} \left (x \sqrt {x^2+\sqrt {1+x^4}}-\sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \] Input:
Integrate[((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
Output:
(x*Sqrt[x^2 + Sqrt[1 + x^4]] - Sqrt[2]*ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/( Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] - 2*Sqrt[2]*ArcTanh[(-1 + x^2 + Sqrt [1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
\(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arcsin \left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}+\frac {i \text {arcsinh}\left (\sqrt [4]{-1} x\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}}+\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {1-i x^2} x+\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {1+i x^2} x\) |
Input:
Int[((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
Output:
(1/4 + I/4)*x*Sqrt[1 - I*x^2] + (1/4 - I/4)*x*Sqrt[1 + I*x^2] - ArcSin[(-1 )^(1/4)*x]/(2*Sqrt[2]) + ((I/2)*ArcSinh[(-1)^(1/4)*x])/Sqrt[2] - ArcTanh[( Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]
\[\int \frac {\left (x^{2}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}}d x\]
Input:
int((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Output:
int((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\text {Timed out} \] Input:
integrate((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="fr icas")
Output:
Timed out
Time = 2.49 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=- \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {1}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} + \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 1 \\\frac {3}{4}, \frac {5}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \] Input:
integrate((x**2-1)*(x**2+(x**4+1)**(1/2))**(1/2)/(x**4+1)**(1/2),x)
Output:
-meijerg(((1, 1), (1/2,)), ((1/4, 3/4), (0,)), x**4)/(4*sqrt(pi)) + meijer g(((3/2, 1), (1,)), ((3/4, 5/4), (0,)), x**4)/(4*sqrt(pi))
\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="ma xima")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))*(x^2 - 1)/sqrt(x^4 + 1), x)
\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="gi ac")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))*(x^2 - 1)/sqrt(x^4 + 1), x)
Timed out. \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \,d x \] Input:
int(((x^2 - 1)*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2),x)
Output:
int(((x^2 - 1)*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2), x)
\[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )}{4}-\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )}{4}+\sqrt {2}\, i +\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{2}}{x^{4}+1}d x \] Input:
int((x^2-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Output:
(sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) - sqrt(2)*x) - sqrt(2)*log(sqrt(s qrt(x**4 + 1) + x**2) + sqrt(2)*x) + 4*sqrt(2)*i + 4*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**2)/(x**4 + 1),x))/4