\(\int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx\) [2516]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \] Output:

-1/2*arctan(2^(1/2)*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2) 
))*2^(1/2)/a^(1/4)/b^(1/4)+1/4*arctanh((2-2^(1/2))*a^(1/4)*b^(1/4)*x/(b^(1 
/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/2)/a^(1/4)/b^(1/4)-1/4*arctanh((2+2 
^(1/2))*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/2)/a 
^(1/4)/b^(1/4)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \] Input:

Integrate[Sqrt[b + a*x^4]/(-b + a*x^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {920, 756, 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.

\(\displaystyle \int \frac {\sqrt {a x^4+b}}{a x^4-b} \, dx\)

\(\Big \downarrow \) 920

\(\displaystyle -\int \frac {1}{1-\frac {4 a b x^4}{\left (a x^4+b\right )^2}}d\frac {x}{\sqrt {a x^4+b}}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {1}{2} \int \frac {1}{1-\frac {2 \sqrt {a} \sqrt {b} x^2}{a x^4+b}}d\frac {x}{\sqrt {a x^4+b}}-\frac {1}{2} \int \frac {1}{\frac {2 \sqrt {a} \sqrt {b} x^2}{a x^4+b}+1}d\frac {x}{\sqrt {a x^4+b}}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {1}{2} \int \frac {1}{1-\frac {2 \sqrt {a} \sqrt {b} x^2}{a x^4+b}}d\frac {x}{\sqrt {a x^4+b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}\)

Input:

Int[Sqrt[b + a*x^4]/(-b + a*x^4),x]
 

Output:

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

Defintions of rubi rules used

rule 216

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])
 

rule 219

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])
 

rule 756

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

rule 920

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.41

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (-2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )+\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}\right )\right )}{8 \left (a b \right )^{\frac {1}{4}}}\)	\(86\)
default	\(\frac {\left (2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{8 \left (a b \right )^{\frac {1}{4}}}\)	\(91\)
elliptic	\(\frac {\left (2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{8 \left (a b \right )^{\frac {1}{4}}}\)	\(91\)

Input:

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

Output:

-1/8*2^(1/2)*(-2*arctan(1/2*(a*x^4+b)^(1/2)*2^(1/2)/x/(a*b)^(1/4))+ln((-2^ 
(1/2)*(a*b)^(1/4)*x-(a*x^4+b)^(1/2))/(2^(1/2)*(a*b)^(1/4)*x-(a*x^4+b)^(1/2 
))))/(a*b)^(1/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} - 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} - b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} - b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) \] Input:

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

Output:

-1/4*(1/4)^(1/4)*(1/(a*b))^(1/4)*log((4*(1/4)^(3/4)*a*b*x^3*(1/(a*b))^(3/4 
) + 2*(1/4)^(1/4)*b*x*(1/(a*b))^(1/4) + sqrt(a*x^4 + b)*(x^2 + b*sqrt(1/(a 
*b))))/(a*x^4 - b)) + 1/4*(1/4)^(1/4)*(1/(a*b))^(1/4)*log(-(4*(1/4)^(3/4)* 
a*b*x^3*(1/(a*b))^(3/4) + 2*(1/4)^(1/4)*b*x*(1/(a*b))^(1/4) - sqrt(a*x^4 + 
 b)*(x^2 + b*sqrt(1/(a*b))))/(a*x^4 - b)) + 1/4*I*(1/4)^(1/4)*(1/(a*b))^(1 
/4)*log((4*I*(1/4)^(3/4)*a*b*x^3*(1/(a*b))^(3/4) - 2*I*(1/4)^(1/4)*b*x*(1/ 
(a*b))^(1/4) + sqrt(a*x^4 + b)*(x^2 - b*sqrt(1/(a*b))))/(a*x^4 - b)) - 1/4 
*I*(1/4)^(1/4)*(1/(a*b))^(1/4)*log((-4*I*(1/4)^(3/4)*a*b*x^3*(1/(a*b))^(3/ 
4) + 2*I*(1/4)^(1/4)*b*x*(1/(a*b))^(1/4) + sqrt(a*x^4 + b)*(x^2 - b*sqrt(1 
/(a*b))))/(a*x^4 - b))
 

Sympy [F]

\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int \frac {\sqrt {a x^{4} + b}}{a x^{4} - b}\, dx \] Input:

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

Output:

Integral(sqrt(a*x**4 + b)/(a*x**4 - b), x)
 

Maxima [F]

\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int { \frac {\sqrt {a x^{4} + b}}{a x^{4} - b} \,d x } \] Input:

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

Output:

integrate(sqrt(a*x^4 + b)/(a*x^4 - b), x)
 

Giac [F]

\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int { \frac {\sqrt {a x^{4} + b}}{a x^{4} - b} \,d x } \] Input:

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

Output:

integrate(sqrt(a*x^4 + b)/(a*x^4 - b), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\int \frac {\sqrt {a\,x^4+b}}{b-a\,x^4} \,d x \] Input:

int(-(b + a*x^4)^(1/2)/(b - a*x^4),x)
 

Output:

-int((b + a*x^4)^(1/2)/(b - a*x^4), x)
 

Reduce [F]

\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int \frac {\sqrt {a \,x^{4}+b}}{a \,x^{4}-b}d x \] Input:

int((a*x^4+b)^(1/2)/(a*x^4-b),x)
 

Output:

int(sqrt(a*x**4 + b)/(a*x**4 - b),x)