\(\int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx\) [471]

Optimal result

Integrand size = 20, antiderivative size = 22 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=\arctan \left (\frac {x}{2}\right )-\frac {3 \arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]

[Out]

arctan(1/2*x)-3/2*arctan(x*2^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1180, 209} \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=\arctan \left (\frac {x}{2}\right )-\frac {3 \arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]

[In]

Int[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

ArcTan[x/2] - (3*ArcTan[Sqrt[2]*x])/Sqrt[2]

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 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\left (3 \int \frac {1}{1+2 x^2} \, dx\right )+4 \int \frac {1}{8+2 x^2} \, dx \\ & = \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3 \tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=\arctan \left (\frac {x}{2}\right )-\frac {3 \arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]

[In]

Integrate[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

ArcTan[x/2] - (3*ArcTan[Sqrt[2]*x])/Sqrt[2]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
default	\(\arctan \left (\frac {x}{2}\right )-\frac {3 \arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\)	\(17\)
risch	\(\arctan \left (\frac {x}{2}\right )-\frac {3 \arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\)	\(17\)

[In]

int((x^2-10)/(2*x^4+9*x^2+4),x,method=_RETURNVERBOSE)

[Out]

arctan(1/2*x)-3/2*arctan(x*2^(1/2))*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="fricas")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=\operatorname {atan}{\left (\frac {x}{2} \right )} - \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \]

[In]

integrate((x**2-10)/(2*x**4+9*x**2+4),x)

[Out]

atan(x/2) - 3*sqrt(2)*atan(sqrt(2)*x)/2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="maxima")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=-\frac {3}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \arctan \left (\frac {1}{2} \, x\right ) \]

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="giac")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-10+x^2}{4+9 x^2+2 x^4} \, dx=\mathrm {atan}\left (\frac {x}{2}\right )-\frac {3\,\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2} \]

[In]

int((x^2 - 10)/(9*x^2 + 2*x^4 + 4),x)

[Out]

atan(x/2) - (3*2^(1/2)*atan(2^(1/2)*x))/2