Integrand size = 30, antiderivative size = 49 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=-\frac {4}{15 x^3 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}-\frac {1}{5 x^5 \sqrt {x^2+\sqrt {1+x^4}}} \] Output:
-4/15/x^3/(x^2+(x^4+1)^(1/2))^(3/2)-1/5/x^5/(x^2+(x^4+1)^(1/2))^(1/2)
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=\frac {-3-10 x^4-10 x^2 \sqrt {1+x^4}}{15 x^5 \left (x^2+\sqrt {1+x^4}\right )^{5/2}} \] Input:
Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(x^6*Sqrt[1 + x^4]),x]
Output:
(-3 - 10*x^4 - 10*x^2*Sqrt[1 + x^4])/(15*x^5*(x^2 + Sqrt[1 + x^4])^(5/2))
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 157, normalized size of antiderivative = 3.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2558, 245, 245, 242}
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^6 \sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 2558 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{x^6 \sqrt {1-i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{x^6 \sqrt {i x^2+1}}dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {4}{5} i \int \frac {1}{x^4 \sqrt {1-i x^2}}dx-\frac {\sqrt {1-i x^2}}{5 x^5}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {4}{5} i \int \frac {1}{x^4 \sqrt {i x^2+1}}dx-\frac {\sqrt {1+i x^2}}{5 x^5}\right )\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {4}{5} i \left (\frac {2}{3} i \int \frac {1}{x^2 \sqrt {1-i x^2}}dx-\frac {\sqrt {1-i x^2}}{3 x^3}\right )-\frac {\sqrt {1-i x^2}}{5 x^5}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {4}{5} i \left (-\frac {2}{3} i \int \frac {1}{x^2 \sqrt {i x^2+1}}dx-\frac {\sqrt {1+i x^2}}{3 x^3}\right )-\frac {\sqrt {1+i x^2}}{5 x^5}\right )\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {4}{5} i \left (-\frac {2 i \sqrt {1-i x^2}}{3 x}-\frac {\sqrt {1-i x^2}}{3 x^3}\right )-\frac {\sqrt {1-i x^2}}{5 x^5}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {\sqrt {1+i x^2}}{5 x^5}-\frac {4}{5} i \left (\frac {2 i \sqrt {1+i x^2}}{3 x}-\frac {\sqrt {1+i x^2}}{3 x^3}\right )\right )\) |
Input:
Int[Sqrt[x^2 + Sqrt[1 + x^4]]/(x^6*Sqrt[1 + x^4]),x]
Output:
(1/2 - I/2)*(-1/5*Sqrt[1 - I*x^2]/x^5 + ((4*I)/5)*(-1/3*Sqrt[1 - I*x^2]/x^ 3 - (((2*I)/3)*Sqrt[1 - I*x^2])/x)) + (1/2 + I/2)*(-1/5*Sqrt[1 + I*x^2]/x^ 5 - ((4*I)/5)*(-1/3*Sqrt[1 + I*x^2]/x^3 + (((2*I)/3)*Sqrt[1 + I*x^2])/x))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2 Int[(c + d*x) ^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2 Int[(c + d*x)^m/Sqrt[ Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G tQ[a, 0]
\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{6} \sqrt {x^{4}+1}}d x\]
Input:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x)
Output:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=-\frac {{\left (8 \, x^{6} + x^{2} - {\left (8 \, x^{4} - 3\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{15 \, x^{5}} \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x, algorithm="fricas ")
Output:
-1/15*(8*x^6 + x^2 - (8*x^4 - 3)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/ x^5
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (44) = 88\).
Time = 1.46 (sec) , antiderivative size = 298, normalized size of antiderivative = 6.08 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=- \frac {10 \sqrt {2} x^{4} \sqrt {1 + \frac {1}{x^{4}}} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{60 \pi x^{10} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 60 \pi x^{10} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 30 \pi x^{6} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1}} - \frac {10 \sqrt {2} x^{4} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{60 \pi x^{10} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 60 \pi x^{10} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 30 \pi x^{6} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1}} - \frac {3 \sqrt {2} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{60 \pi x^{10} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 60 \pi x^{10} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} + 30 \pi x^{6} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1}} \] Input:
integrate((x**2+(x**4+1)**(1/2))**(1/2)/x**6/(x**4+1)**(1/2),x)
Output:
-10*sqrt(2)*x**4*sqrt(1 + x**(-4))*gamma(1/4)*gamma(3/4)/(60*pi*x**10*sqrt (1 + x**(-4))*sqrt(sqrt(1 + x**(-4)) + 1) + 60*pi*x**10*sqrt(sqrt(1 + x**( -4)) + 1) + 30*pi*x**6*sqrt(sqrt(1 + x**(-4)) + 1)) - 10*sqrt(2)*x**4*gamm a(1/4)*gamma(3/4)/(60*pi*x**10*sqrt(1 + x**(-4))*sqrt(sqrt(1 + x**(-4)) + 1) + 60*pi*x**10*sqrt(sqrt(1 + x**(-4)) + 1) + 30*pi*x**6*sqrt(sqrt(1 + x* *(-4)) + 1)) - 3*sqrt(2)*gamma(1/4)*gamma(3/4)/(60*pi*x**10*sqrt(1 + x**(- 4))*sqrt(sqrt(1 + x**(-4)) + 1) + 60*pi*x**10*sqrt(sqrt(1 + x**(-4)) + 1) + 30*pi*x**6*sqrt(sqrt(1 + x**(-4)) + 1))
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} x^{6}} \,d x } \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x, algorithm="maxima ")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))/(sqrt(x^4 + 1)*x^6), x)
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} x^{6}} \,d x } \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))/(sqrt(x^4 + 1)*x^6), x)
Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^6\,\sqrt {x^4+1}} \,d x \] Input:
int(((x^4 + 1)^(1/2) + x^2)^(1/2)/(x^6*(x^4 + 1)^(1/2)),x)
Output:
int(((x^4 + 1)^(1/2) + x^2)^(1/2)/(x^6*(x^4 + 1)^(1/2)), x)
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^6 \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{10}+x^{6}}d x \] Input:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^6/(x^4+1)^(1/2),x)
Output:
int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**10 + x**6),x)