Integrand size = 23, antiderivative size = 210 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \]
-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)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \]
-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))
Time = 0.20 (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 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}\) |
-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))
3.26.16.3.1 Defintions of rubi rules used
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[((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]
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]
Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.42
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )-\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}\right )\right )}{8 \left (a b \right )^{\frac {1}{4}}}\) | \(88\) |
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 | \(\frac {\left (2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{8 \left (a b \right )^{\frac {1}{4}}}\) | \(91\) |
1/8*2^(1/2)/(a*b)^(1/4)*(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))))
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} - 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} - b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} - b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) \]
-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))
\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int \frac {\sqrt {a x^{4} + b}}{a x^{4} - b}\, dx \]
\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int { \frac {\sqrt {a x^{4} + b}}{a x^{4} - b} \,d x } \]
\[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=\int { \frac {\sqrt {a x^{4} + b}}{a x^{4} - b} \,d x } \]
Timed out. \[ \int \frac {\sqrt {b+a x^4}}{-b+a x^4} \, dx=-\int \frac {\sqrt {a\,x^4+b}}{b-a\,x^4} \,d x \]