Integrand size = 30, antiderivative size = 73 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=-\frac {1}{7 x^7 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {6 \sqrt {x^2+\sqrt {1+x^4}}}{35 x^5}+\frac {8 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}{35 x^3} \] Output:
-1/7/x^7/(x^2+(x^4+1)^(1/2))^(1/2)-6/35*(x^2+(x^4+1)^(1/2))^(1/2)/x^5+8/35 *(x^2+(x^4+1)^(1/2))^(3/2)/x^3
Time = 0.53 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4} \left (-5-36 x^4+48 x^8+128 x^{12}\right )+x^2 \left (-21-28 x^4+112 x^8+128 x^{12}\right )}{35 x^7 \left (x^2+\sqrt {1+x^4}\right )^{7/2}} \] Input:
Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(x^8*Sqrt[1 + x^4]),x]
Output:
(Sqrt[1 + x^4]*(-5 - 36*x^4 + 48*x^8 + 128*x^12) + x^2*(-21 - 28*x^4 + 112 *x^8 + 128*x^12))/(35*x^7*(x^2 + Sqrt[1 + x^4])^(7/2))
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2558, 245, 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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^8 \sqrt {x^4+1}} \, dx\) |
\(\Big \downarrow \) 2558 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{x^8 \sqrt {1-i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{x^8 \sqrt {i x^2+1}}dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {6}{7} i \int \frac {1}{x^6 \sqrt {1-i x^2}}dx-\frac {\sqrt {1-i x^2}}{7 x^7}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {6}{7} i \int \frac {1}{x^6 \sqrt {i x^2+1}}dx-\frac {\sqrt {1+i x^2}}{7 x^7}\right )\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {6}{7} i \left (\frac {4}{5} i \int \frac {1}{x^4 \sqrt {1-i x^2}}dx-\frac {\sqrt {1-i x^2}}{5 x^5}\right )-\frac {\sqrt {1-i x^2}}{7 x^7}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {6}{7} i \left (-\frac {4}{5} i \int \frac {1}{x^4 \sqrt {i x^2+1}}dx-\frac {\sqrt {1+i x^2}}{5 x^5}\right )-\frac {\sqrt {1+i x^2}}{7 x^7}\right )\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {6}{7} i \left (\frac {4}{5} i \left (\frac {2}{3} i \int \frac {1}{x^2 \sqrt {1-i x^2}}dx-\frac {\sqrt {1-i x^2}}{3 x^3}\right )-\frac {\sqrt {1-i x^2}}{5 x^5}\right )-\frac {\sqrt {1-i x^2}}{7 x^7}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {6}{7} i \left (-\frac {4}{5} i \left (-\frac {2}{3} i \int \frac {1}{x^2 \sqrt {i x^2+1}}dx-\frac {\sqrt {1+i x^2}}{3 x^3}\right )-\frac {\sqrt {1+i x^2}}{5 x^5}\right )-\frac {\sqrt {1+i x^2}}{7 x^7}\right )\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {6}{7} i \left (\frac {4}{5} i \left (-\frac {2 i \sqrt {1-i x^2}}{3 x}-\frac {\sqrt {1-i x^2}}{3 x^3}\right )-\frac {\sqrt {1-i x^2}}{5 x^5}\right )-\frac {\sqrt {1-i x^2}}{7 x^7}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {\sqrt {1+i x^2}}{7 x^7}-\frac {6}{7} i \left (-\frac {\sqrt {1+i x^2}}{5 x^5}-\frac {4}{5} i \left (\frac {2 i \sqrt {1+i x^2}}{3 x}-\frac {\sqrt {1+i x^2}}{3 x^3}\right )\right )\right )\) |
Input:
Int[Sqrt[x^2 + Sqrt[1 + x^4]]/(x^8*Sqrt[1 + x^4]),x]
Output:
(1/2 - I/2)*(-1/7*Sqrt[1 - I*x^2]/x^7 + ((6*I)/7)*(-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)*(-1/7*Sqrt[1 + I*x^2]/x^7 - ((6*I)/7)*(-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]
\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{8} \sqrt {x^{4}+1}}d x\]
Input:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x)
Output:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\frac {{\left (8 \, x^{6} - x^{2} + {\left (8 \, x^{4} - 5\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{35 \, x^{7}} \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x, algorithm="fricas ")
Output:
1/35*(8*x^6 - x^2 + (8*x^4 - 5)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x ^7
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (65) = 130\).
Time = 1.80 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.49 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\frac {3 \sqrt {2} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} \Gamma \left (- \frac {7}{4}\right ) \Gamma \left (- \frac {5}{4}\right )}{64 \pi } + \frac {3 \sqrt {2} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} \Gamma \left (- \frac {7}{4}\right ) \Gamma \left (- \frac {5}{4}\right )}{64 \pi } - \frac {15 \sqrt {2} \sqrt {1 + \frac {1}{x^{4}}} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} \Gamma \left (- \frac {7}{4}\right ) \Gamma \left (- \frac {5}{4}\right )}{512 \pi x^{4}} - \frac {3 \sqrt {2} \sqrt {\sqrt {1 + \frac {1}{x^{4}}} + 1} \Gamma \left (- \frac {7}{4}\right ) \Gamma \left (- \frac {5}{4}\right )}{512 \pi x^{4}} \] Input:
integrate((x**2+(x**4+1)**(1/2))**(1/2)/x**8/(x**4+1)**(1/2),x)
Output:
3*sqrt(2)*sqrt(1 + x**(-4))*sqrt(sqrt(1 + x**(-4)) + 1)*gamma(-7/4)*gamma( -5/4)/(64*pi) + 3*sqrt(2)*sqrt(sqrt(1 + x**(-4)) + 1)*gamma(-7/4)*gamma(-5 /4)/(64*pi) - 15*sqrt(2)*sqrt(1 + x**(-4))*sqrt(sqrt(1 + x**(-4)) + 1)*gam ma(-7/4)*gamma(-5/4)/(512*pi*x**4) - 3*sqrt(2)*sqrt(sqrt(1 + x**(-4)) + 1) *gamma(-7/4)*gamma(-5/4)/(512*pi*x**4)
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} x^{8}} \,d x } \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x, algorithm="maxima ")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))/(sqrt(x^4 + 1)*x^8), x)
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} x^{8}} \,d x } \] Input:
integrate((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 + sqrt(x^4 + 1))/(sqrt(x^4 + 1)*x^8), x)
Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^8\,\sqrt {x^4+1}} \,d x \] Input:
int(((x^4 + 1)^(1/2) + x^2)^(1/2)/(x^8*(x^4 + 1)^(1/2)),x)
Output:
int(((x^4 + 1)^(1/2) + x^2)^(1/2)/(x^8*(x^4 + 1)^(1/2)), x)
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^8 \sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{12}+x^{8}}d x \] Input:
int((x^2+(x^4+1)^(1/2))^(1/2)/x^8/(x^4+1)^(1/2),x)
Output:
int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**12 + x**8),x)