\(\int \frac {1}{\sqrt [4]{-b+a x^4} (b+a x^4)} \, dx\) [1185]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

[Out]

1/4*arctan(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))*2^(3/4)/a^(1/4)/b+1/4*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4-b)^(1/4))
*2^(3/4)/a^(1/4)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {385, 218, 212, 209} \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

[In]

Int[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]

[Out]

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

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

[In]

Integrate[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]

[Out]

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

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2^{\frac {1}{4}} a^{\frac {1}{4}} x -\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )\right )}{8 a^{\frac {1}{4}} b}\)	\(87\)

[In]

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

[Out]

1/8*2^(3/4)*(-2*arctan(1/2*2^(3/4)/a^(1/4)/x*(a*x^4-b)^(1/4))+ln((-2^(1/4)*a^(1/4)*x-(a*x^4-b)^(1/4))/(2^(1/4)
*a^(1/4)*x-(a*x^4-b)^(1/4))))/a^(1/4)/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 106.34 (sec) , antiderivative size = 525, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) + \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (-3 i \, a b x^{4} + i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} i \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {-4 i \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 i \, a b x^{4} - i \, b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) \]

[In]

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

[Out]

1/8*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(1/2*(4*(1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(
1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) + 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*(3*a*b*x^4 - b^2)*(1/(a
*b^4))^(1/4))/(a*x^4 + b)) - 1/8*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-1/2*(4*(1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x
^2*(1/(a*b^4))^(3/4) - 4*sqrt(1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 - b)^(3/4)*x + (1/2)
^(1/4)*(3*a*b*x^4 - b^2)*(1/(a*b^4))^(1/4))/(a*x^4 + b)) + 1/8*I*(1/2)^(1/4)*(1/(a*b^4))^(1/4)*log(-1/2*(4*I*(
1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4
)) - 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*(-3*I*a*b*x^4 + I*b^2)*(1/(a*b^4))^(1/4))/(a*x^4 + b)) - 1/8*I*(1/2)^
(1/4)*(1/(a*b^4))^(1/4)*log(-1/2*(-4*I*(1/2)^(3/4)*sqrt(a*x^4 - b)*a*b^3*x^2*(1/(a*b^4))^(3/4) + 4*sqrt(1/2)*(
a*x^4 - b)^(1/4)*a*b^2*x^3*sqrt(1/(a*b^4)) - 2*(a*x^4 - b)^(3/4)*x + (1/2)^(1/4)*(3*I*a*b*x^4 - I*b^2)*(1/(a*b
^4))^(1/4))/(a*x^4 + b))

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{4} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx=\int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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