\(\int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} (-b+3 a x^4)} \, dx\) [1914]

Optimal result

Integrand size = 33, antiderivative size = 133 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]

[Out]

1/6*arctan(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)+1/6*arctan(2^(1/2)*a^(1/4)*x/(a*x^4+b)^(1/4))*2^(1/2)/a^(1/4)+1/
6*arctanh(a^(1/4)*x/(a*x^4+b)^(1/4))/a^(1/4)+1/6*arctanh(2^(1/2)*a^(1/4)*x/(a*x^4+b)^(1/4))*2^(1/2)/a^(1/4)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {544, 246, 218, 212, 209, 385} \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]

[In]

Int[(-b + a*x^4)/((b + a*x^4)^(1/4)*(-b + 3*a*x^4)),x]

[Out]

ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTan[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)]/(3*Sqrt[2]*a
^(1/4)) + ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTanh[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)]/
(3*Sqrt[2]*a^(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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx-\frac {1}{3} (2 b) \int \frac {1}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{3} (2 b) \text {Subst}\left (\int \frac {1}{-b+4 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-2 \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+2 \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{6 \sqrt [4]{a}} \]

[In]

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

[Out]

(ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)] + ArcTanh[(a^(1
/4)*x)/(b + a*x^4)^(1/4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*x)/(b + a*x^4)^(1/4)])/(6*a^(1/4))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right ) \sqrt {2}+\ln \left (\frac {x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}} \sqrt {2}+\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )}{12 a^{\frac {1}{4}}}\)	\(136\)

[In]

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

[Out]

1/12*(-2*arctan(1/2*2^(1/2)/a^(1/4)/x*(a*x^4+b)^(1/4))*2^(1/2)+ln((x*a^(1/4)*2^(1/2)+(a*x^4+b)^(1/4))/(-x*a^(1
/4)*2^(1/2)+(a*x^4+b)^(1/4)))*2^(1/2)-2*arctan(1/a^(1/4)/x*(a*x^4+b)^(1/4))+ln((-a^(1/4)*x-(a*x^4+b)^(1/4))/(a
^(1/4)*x-(a*x^4+b)^(1/4))))/a^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.78 \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} \]

[In]

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

[Out]

1/6*(1/4)^(1/4)*log((4*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/6*(1/4)^(1/4)*log(-(4*(1/4)^(
3/4)*a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/6*I*(1/4)^(1/4)*log((4*I*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 + b
)^(1/4))/x)/a^(1/4) + 1/6*I*(1/4)^(1/4)*log((-4*I*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) + 1/12
*log((a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/12*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/12*
I*log((I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) + 1/12*I*log((-I*a^(1/4)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4)

Sympy [F]

\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {a x^{4} - b}{\sqrt [4]{a x^{4} + b} \left (3 a x^{4} - b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (3 \, a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (-b+3 a x^4\right )} \, dx=\int \frac {b-a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (b-3\,a\,x^4\right )} \,d x \]

[In]

int((b - a*x^4)/((b + a*x^4)^(1/4)*(b - 3*a*x^4)),x)

[Out]

int((b - a*x^4)/((b + a*x^4)^(1/4)*(b - 3*a*x^4)), x)