\(\int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx\) [1239]

Optimal result

Integrand size = 21, antiderivative size = 90 \[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {1+3 x^4}}{1-\sqrt {2} \sqrt [4]{3} x+\sqrt {3} x^2}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]

[Out]

-1/6*arctan((3*x^4+1)^(1/2)/(1-2^(1/2)*3^(1/4)*x+3^(1/2)*x^2))*2^(1/2)*3^(3/4)-1/12*arctanh(2^(1/2)*3^(1/4)*x/
(3*x^4+1)^(1/2))*2^(1/2)*3^(3/4)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {413, 218, 212, 209} \[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3 x^4+1}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3 x^4+1}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]

[In]

Int[Sqrt[1 + 3*x^4]/(-1 + 3*x^4),x]

[Out]

-1/2*ArcTan[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^4]]/(Sqrt[2]*3^(1/4)) - ArcTanh[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^
4]]/(2*Sqrt[2]*3^(1/4))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 413

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[a/c, Subst[Int[1/(1 - 4*a*b*x^4), x], x
, x/Sqrt[a + b*x^4]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && PosQ[a*b]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1-12 x^4} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-2 \sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right )\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+2 \sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt {1+3 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {1+3 x^4}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]

[In]

Integrate[Sqrt[1 + 3*x^4]/(-1 + 3*x^4),x]

[Out]

-1/2*(ArcTan[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^4]] + ArcTanh[(Sqrt[2]*3^(1/4)*x)/Sqrt[1 + 3*x^4]])/(Sqrt[2]*3^(
1/4))

Maple [A] (verified)

Time = 6.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {3^{\frac {3}{4}} \left (2 \arctan \left (\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}\, 3^{\frac {3}{4}}}{6 x}\right )-\ln \left (\frac {\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}+3^{\frac {1}{4}}}{\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}-3^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{24}\)	\(83\)
elliptic	\(\frac {3^{\frac {3}{4}} \left (2 \arctan \left (\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}\, 3^{\frac {3}{4}}}{6 x}\right )-\ln \left (\frac {\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}+3^{\frac {1}{4}}}{\frac {\sqrt {3 x^{4}+1}\, \sqrt {2}}{2 x}-3^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{24}\)	\(83\)
pseudoelliptic	\(-\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} x}{\sqrt {3 x^{4}+1}}\right )+\operatorname {arctanh}\left (\frac {\left (3^{\frac {3}{4}} x^{2}-x \sqrt {3}+3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}} \sqrt {2}}{3 \sqrt {3 x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (3^{\frac {3}{4}} x^{2}+x \sqrt {3}+3^{\frac {1}{4}}\right ) 3^{\frac {3}{4}} \sqrt {2}}{3 \sqrt {3 x^{4}+1}}\right )\right )}{24}\)	\(101\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right ) \ln \left (\frac {x \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{3}+18 \sqrt {3 x^{4}+1}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} x^{2}-6}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (\frac {x \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right )+18 \sqrt {3 x^{4}+1}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} x^{2}+6}\right )}{12}\)	\(118\)

[In]

int((3*x^4+1)^(1/2)/(3*x^4-1),x,method=_RETURNVERBOSE)

[Out]

1/24*3^(3/4)*(2*arctan(1/6*(3*x^4+1)^(1/2)*2^(1/2)/x*3^(3/4))-ln((1/2*(3*x^4+1)^(1/2)*2^(1/2)/x+3^(1/4))/(1/2*
(3*x^4+1)^(1/2)*2^(1/2)/x-3^(1/4))))*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=-\frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (\frac {6 \cdot 12^{\frac {1}{4}} x^{3} + 12^{\frac {3}{4}} x + 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} + \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) + \frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (-\frac {6 \cdot 12^{\frac {1}{4}} x^{3} + 12^{\frac {3}{4}} x - 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} + \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) + \frac {1}{48} i \cdot 12^{\frac {3}{4}} \log \left (\frac {6 i \cdot 12^{\frac {1}{4}} x^{3} - i \cdot 12^{\frac {3}{4}} x + 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} - \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) - \frac {1}{48} i \cdot 12^{\frac {3}{4}} \log \left (\frac {-6 i \cdot 12^{\frac {1}{4}} x^{3} + i \cdot 12^{\frac {3}{4}} x + 2 \, \sqrt {3 \, x^{4} + 1} {\left (3 \, x^{2} - \sqrt {3}\right )}}{3 \, x^{4} - 1}\right ) \]

[In]

integrate((3*x^4+1)^(1/2)/(3*x^4-1),x, algorithm="fricas")

[Out]

-1/48*12^(3/4)*log((6*12^(1/4)*x^3 + 12^(3/4)*x + 2*sqrt(3*x^4 + 1)*(3*x^2 + sqrt(3)))/(3*x^4 - 1)) + 1/48*12^
(3/4)*log(-(6*12^(1/4)*x^3 + 12^(3/4)*x - 2*sqrt(3*x^4 + 1)*(3*x^2 + sqrt(3)))/(3*x^4 - 1)) + 1/48*I*12^(3/4)*
log((6*I*12^(1/4)*x^3 - I*12^(3/4)*x + 2*sqrt(3*x^4 + 1)*(3*x^2 - sqrt(3)))/(3*x^4 - 1)) - 1/48*I*12^(3/4)*log
((-6*I*12^(1/4)*x^3 + I*12^(3/4)*x + 2*sqrt(3*x^4 + 1)*(3*x^2 - sqrt(3)))/(3*x^4 - 1))

Sympy [F]

\[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=\int \frac {\sqrt {3 x^{4} + 1}}{3 x^{4} - 1}\, dx \]

[In]

integrate((3*x**4+1)**(1/2)/(3*x**4-1),x)

[Out]

Integral(sqrt(3*x**4 + 1)/(3*x**4 - 1), x)

Maxima [F]

\[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 1}}{3 \, x^{4} - 1} \,d x } \]

[In]

integrate((3*x^4+1)^(1/2)/(3*x^4-1),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x^4 + 1)/(3*x^4 - 1), x)

Giac [F]

\[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 1}}{3 \, x^{4} - 1} \,d x } \]

[In]

integrate((3*x^4+1)^(1/2)/(3*x^4-1),x, algorithm="giac")

[Out]

integrate(sqrt(3*x^4 + 1)/(3*x^4 - 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1+3 x^4}}{-1+3 x^4} \, dx=\int \frac {\sqrt {3\,x^4+1}}{3\,x^4-1} \,d x \]

[In]

int((3*x^4 + 1)^(1/2)/(3*x^4 - 1),x)

[Out]

int((3*x^4 + 1)^(1/2)/(3*x^4 - 1), x)