Integrand size = 32, antiderivative size = 101 \[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {2 x^4 \sqrt {1+x^4}+x^2 \left (3+2 x^4\right )}{8 x \sqrt {x^2+\sqrt {1+x^4}}}-\frac {11 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{4 \sqrt {2}} \] Output:
1/8*(2*x^4*(x^4+1)^(1/2)+x^2*(2*x^4+3))/x/(x^2+(x^4+1)^(1/2))^(1/2)-11/8*a rctanh(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 = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {3 x+2 x^5+2 x^3 \sqrt {1+x^4}-11 \sqrt {2} \sqrt {x^2+\sqrt {1+x^4}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{8 \sqrt {x^2+\sqrt {1+x^4}}} \] Input:
Integrate[((-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
Output:
(3*x + 2*x^5 + 2*x^3*Sqrt[1 + x^4] - 11*Sqrt[2]*Sqrt[x^2 + Sqrt[1 + x^4]]* ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]) /(8*Sqrt[x^2 + Sqrt[1 + x^4]])
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59, 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.
\(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^4 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}-\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 i \arcsin \left (\sqrt [4]{-1} x\right )}{8 \sqrt {2}}-\frac {3 \text {arcsinh}\left (\sqrt [4]{-1} x\right )}{8 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}}+\left (\frac {3}{16}-\frac {3 i}{16}\right ) \sqrt {1-i x^2} x+\left (\frac {3}{16}+\frac {3 i}{16}\right ) \sqrt {1+i x^2} x+\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt {1-i x^2} x^3+\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {1+i x^2} x^3\) |
Input:
Int[((-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[1 + x^4],x]
Output:
(3/16 - (3*I)/16)*x*Sqrt[1 - I*x^2] + (1/8 + I/8)*x^3*Sqrt[1 - I*x^2] + (3 /16 + (3*I)/16)*x*Sqrt[1 + I*x^2] + (1/8 - I/8)*x^3*Sqrt[1 + I*x^2] + (((3 *I)/8)*ArcSin[(-1)^(1/4)*x])/Sqrt[2] - (3*ArcSinh[(-1)^(1/4)*x])/(8*Sqrt[2 ]) - ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]
\[\int \frac {\left (x^{4}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}}d x\]
Input:
int((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Output:
int((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=-\frac {1}{8} \, {\left (x^{3} - 3 \, \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {11}{32} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \] Input:
integrate((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="fr icas")
Output:
-1/8*(x^3 - 3*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 11/32*sqrt(2)*l og(4*x^4 + 4*sqrt(x^4 + 1)*x^2 - 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x) *sqrt(x^2 + sqrt(x^4 + 1)) + 1)
Time = 19.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=- \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {1}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} + \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 2, 1 & \frac {3}{2} \\\frac {5}{4}, \frac {7}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \] Input:
integrate((x**4-1)*(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(((2, 1), (3/2,)), ((5/4, 7/4), (0,)), x**4)/(4*sqrt(pi))
\[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="ma xima")
Output:
integrate((x^4 - 1)*sqrt(x^2 + sqrt(x^4 + 1))/sqrt(x^4 + 1), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}} \,d x } \] Input:
integrate((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x, algorithm="gi ac")
Output:
integrate((x^4 - 1)*sqrt(x^2 + sqrt(x^4 + 1))/sqrt(x^4 + 1), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \,d x \] Input:
int(((x^4 - 1)*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2),x)
Output:
int(((x^4 - 1)*((x^4 + 1)^(1/2) + x^2)^(1/2))/(x^4 + 1)^(1/2), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {11 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )}{32}-\frac {11 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )}{32}+\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{4}}{x^{4}+1}d x +\frac {3 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{4}+1}d x \right )}{8} \] Input:
int((x^4-1)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)
Output:
(11*sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) - sqrt(2)*x) - 11*sqrt(2)*log( sqrt(sqrt(x**4 + 1) + x**2) + sqrt(2)*x) + 32*int((sqrt(sqrt(x**4 + 1) + x **2)*sqrt(x**4 + 1)*x**4)/(x**4 + 1),x) + 12*int((sqrt(sqrt(x**4 + 1) + x* *2)*sqrt(x**4 + 1))/(x**4 + 1),x))/32