Integrand size = 30, antiderivative size = 24 \[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {1}{3 x^3 \left (x^2+\sqrt {1+x^4}\right )^{3/2}} \] Output:
-1/3/x^3/(x^2+(x^4+1)^(1/2))^(3/2)
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {1}{3 x^3 \left (x^2+\sqrt {1+x^4}\right )^{3/2}} \] Input:
Integrate[1/(x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Output:
-1/3*1/(x^3*(x^2 + Sqrt[1 + x^4])^(3/2))
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.
\(\displaystyle \int \frac {1}{x^4 \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {1}{x^4 \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}}dx\) |
Input:
Int[1/(x^4*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Output:
$Aborted
\[\int \frac {1}{x^{4} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}d x\]
Input:
int(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
Output:
int(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {{\left (2 \, x^{4} - 2 \, \sqrt {x^{4} + 1} x^{2} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{3 \, x^{3}} \] Input:
integrate(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fric as")
Output:
-1/3*(2*x^4 - 2*sqrt(x^4 + 1)*x^2 + 1)*sqrt(x^2 + sqrt(x^4 + 1))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.95 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=- \frac {2 \sqrt {2} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right )}{3 \pi x^{6} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 3 \pi x^{6} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1}} \] Input:
integrate(1/x**4/(x**4+1)**(1/2)/(x**2+(x**4+1)**(1/2))**(1/2),x)
Output:
-2*sqrt(2)*gamma(3/4)*gamma(5/4)/(3*pi*x**6*sqrt(1 + x**(-4))*sqrt(sqrt(1 + x**(-4)) + 1) + 3*pi*x**6*sqrt(sqrt(1 + x**(-4)) + 1))
\[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxi ma")
Output:
integrate(1/(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^4), x)
\[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac ")
Output:
integrate(1/(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {1}{x^4\,\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \] Input:
int(1/(x^4*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2)),x)
Output:
int(1/(x^4*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2)), x)
\[ \int \frac {1}{x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}}{x^{8}+x^{4}}d x +\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}}{x^{4}+1}d x -\left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{6}+x^{2}}d x \right ) \] Input:
int(1/x^4/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x)
Output:
int(sqrt(sqrt(x**4 + 1) + x**2)/(x**8 + x**4),x) + int(sqrt(sqrt(x**4 + 1) + x**2)/(x**4 + 1),x) - int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/ (x**6 + x**2),x)