\(\int \frac {-b+a x^2}{(b+a x^2) \sqrt {b^2+a^2 x^4}} \, dx\) [762]

Optimal result

Integrand size = 34, antiderivative size = 59 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{b+a x^2+\sqrt {b^2+a^2 x^4}}\right )}{\sqrt {a} \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1713, 211} \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}} \]

[In]

Int[(-b + a*x^2)/((b + a*x^2)*Sqrt[b^2 + a^2*x^4]),x]

[Out]

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

Rule 211

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

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b}} \]

[In]

Integrate[(-b + a*x^2)/((b + a*x^2)*Sqrt[b^2 + a^2*x^4]),x]

[Out]

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

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64

method	result	size
elliptic	\(\frac {\arctan \left (\frac {\sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {2}}{2 x \sqrt {a b}}\right ) \sqrt {2}}{2 \sqrt {a b}}\)	\(38\)
default	\(-\frac {\sqrt {2}\, \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a b x +\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}\right ) a}{a \,x^{2}+b}\right )\right )}{2 \sqrt {-a b}}\)	\(56\)
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a b x +\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}\right ) a}{a \,x^{2}+b}\right )\right )}{2 \sqrt {-a b}}\)	\(56\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=\left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{2} x^{4} + 2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} a b x \sqrt {-\frac {1}{a b}} - 2 \, a b x^{2} + b^{2}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ), \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {\frac {1}{a b}}}{2 \, x}\right )\right ] \]

[In]

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

[Out]

[1/4*sqrt(2)*sqrt(-1/(a*b))*log((a^2*x^4 + 2*sqrt(2)*sqrt(a^2*x^4 + b^2)*a*b*x*sqrt(-1/(a*b)) - 2*a*b*x^2 + b^
2)/(a^2*x^4 + 2*a*b*x^2 + b^2)), 1/2*sqrt(2)*sqrt(1/(a*b))*arctan(1/2*sqrt(2)*sqrt(a^2*x^4 + b^2)*sqrt(1/(a*b)
)/x)]

Sympy [F]

\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=\int \frac {a x^{2} - b}{\left (a x^{2} + b\right ) \sqrt {a^{2} x^{4} + b^{2}}}\, dx \]

[In]

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

[Out]

Integral((a*x**2 - b)/((a*x**2 + b)*sqrt(a**2*x**4 + b**2)), x)

Maxima [F]

\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=\int { \frac {a x^{2} - b}{\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=\int { \frac {a x^{2} - b}{\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx=\int -\frac {b-a\,x^2}{\sqrt {a^2\,x^4+b^2}\,\left (a\,x^2+b\right )} \,d x \]

[In]

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

[Out]

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