Integrand size = 15, antiderivative size = 68 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}} \] Output:
-(4*x^4+3)^(1/4)/x-1/2*arctan(x*2^(1/2)/(4*x^4+3)^(1/4))*2^(1/2)+1/2*arcta nh(x*2^(1/2)/(4*x^4+3)^(1/4))*2^(1/2)
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}} \] Input:
Integrate[(3 + 4*x^4)^(1/4)/x^2,x]
Output:
-((3 + 4*x^4)^(1/4)/x) - ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)]/Sqrt[2] + A rcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)]/Sqrt[2]
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {809, 854, 827, 216, 219}
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.
\(\displaystyle \int \frac {\sqrt [4]{4 x^4+3}}{x^2} \, dx\) |
\(\Big \downarrow \) 809 |
\(\displaystyle 4 \int \frac {x^2}{\left (4 x^4+3\right )^{3/4}}dx-\frac {\sqrt [4]{4 x^4+3}}{x}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle 4 \int \frac {x^2}{\sqrt {4 x^4+3} \left (1-\frac {4 x^4}{4 x^4+3}\right )}d\frac {x}{\sqrt [4]{4 x^4+3}}-\frac {\sqrt [4]{4 x^4+3}}{x}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 4 \left (\frac {1}{4} \int \frac {1}{1-\frac {2 x^2}{\sqrt {4 x^4+3}}}d\frac {x}{\sqrt [4]{4 x^4+3}}-\frac {1}{4} \int \frac {1}{\frac {2 x^2}{\sqrt {4 x^4+3}}+1}d\frac {x}{\sqrt [4]{4 x^4+3}}\right )-\frac {\sqrt [4]{4 x^4+3}}{x}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 4 \left (\frac {1}{4} \int \frac {1}{1-\frac {2 x^2}{\sqrt {4 x^4+3}}}d\frac {x}{\sqrt [4]{4 x^4+3}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}\right )-\frac {\sqrt [4]{4 x^4+3}}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 4 \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}\right )-\frac {\sqrt [4]{4 x^4+3}}{x}\) |
Input:
Int[(3 + 4*x^4)^(1/4)/x^2,x]
Output:
-((3 + 4*x^4)^(1/4)/x) + 4*(-1/4*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)]/Sqr t[2] + ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)]/(4*Sqrt[2]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 3.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.29
method | result | size |
meijerg | \(-\frac {3^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -\frac {4 x^{4}}{3}\right )}{x}\) | \(20\) |
risch | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {4 \,3^{\frac {1}{4}} x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{9}\) | \(35\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right ) x +\sqrt {2}\, \arctan \left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right ) x -2 \left (4 x^{4}+3\right )^{\frac {1}{4}}}{2 x}\) | \(64\) |
trager | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \sqrt {4 x^{4}+3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )\right )}{4}\) | \(166\) |
Input:
int((4*x^4+3)^(1/4)/x^2,x,method=_RETURNVERBOSE)
Output:
-3^(1/4)/x*hypergeom([-1/4,-1/4],[3/4],-4/3*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (55) = 110\).
Time = 1.97 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {2 \, \sqrt {2} x \arctan \left (\frac {4}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + \frac {2}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x\right ) - \sqrt {2} x \log \left (-256 \, x^{8} - 192 \, x^{4} - 4 \, \sqrt {2} {\left (16 \, x^{5} + 3 \, x\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} - 8 \, \sqrt {2} {\left (16 \, x^{7} + 9 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} - 16 \, {\left (8 \, x^{6} + 3 \, x^{2}\right )} \sqrt {4 \, x^{4} + 3} - 9\right ) + 8 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{8 \, x} \] Input:
integrate((4*x^4+3)^(1/4)/x^2,x, algorithm="fricas")
Output:
-1/8*(2*sqrt(2)*x*arctan(4/3*sqrt(2)*(4*x^4 + 3)^(1/4)*x^3 + 2/3*sqrt(2)*( 4*x^4 + 3)^(3/4)*x) - sqrt(2)*x*log(-256*x^8 - 192*x^4 - 4*sqrt(2)*(16*x^5 + 3*x)*(4*x^4 + 3)^(3/4) - 8*sqrt(2)*(16*x^7 + 9*x^3)*(4*x^4 + 3)^(1/4) - 16*(8*x^6 + 3*x^2)*sqrt(4*x^4 + 3) - 9) + 8*(4*x^4 + 3)^(1/4))/x
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {\sqrt [4]{3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((4*x**4+3)**(1/4)/x**2,x)
Output:
3**(1/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), 4*x**4*exp_polar(I*pi)/3) /(4*x*gamma(3/4))
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \] Input:
integrate((4*x^4+3)^(1/4)/x^2,x, algorithm="maxima")
Output:
1/2*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) - 1/4*sqrt(2)*log(-(sq rt(2) - (4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x)) - (4*x^4 + 3 )^(1/4)/x
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \] Input:
integrate((4*x^4+3)^(1/4)/x^2,x, algorithm="giac")
Output:
1/2*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) - 1/4*sqrt(2)*log(-(sq rt(2) - (4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x)) - (4*x^4 + 3 )^(1/4)/x
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {3^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {4\,x^4}{3}\right )}{x} \] Input:
int((4*x^4 + 3)^(1/4)/x^2,x)
Output:
-(3^(1/4)*hypergeom([-1/4, -1/4], 3/4, -(4*x^4)/3))/x
\[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\int \frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x^{2}}d x \] Input:
int((4*x^4+3)^(1/4)/x^2,x)
Output:
int((4*x**4 + 3)**(1/4)/x**2,x)