\(\int \frac {d+e x^2}{d^2+e^2 x^4} \, dx\) [11]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \]

[Out]

1/2*arctan(-1+x*2^(1/2)*e^(1/2)/d^(1/2))*2^(1/2)/d^(1/2)/e^(1/2)+1/2*arctan(1+x*2^(1/2)*e^(1/2)/d^(1/2))*2^(1/
2)/d^(1/2)/e^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1176, 631, 210} \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \]

[In]

Int[(d + e*x^2)/(d^2 + e^2*x^4),x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]
/(Sqrt[2]*Sqrt[d]*Sqrt[e])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1176

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {-\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )+\arctan \left (1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \]

[In]

Integrate[(d + e*x^2)/(d^2 + e^2*x^4),x]

[Out]

(-ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]] + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[2]*Sqrt[d]*Sqrt[e])

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {\sqrt {2}\, \ln \left (e \,x^{2} \sqrt {-e d}-d e x \sqrt {2}-d \sqrt {-e d}\right )}{4 \sqrt {-e d}}+\frac {\sqrt {2}\, \ln \left (e \,x^{2} \sqrt {-e d}+d e x \sqrt {2}-d \sqrt {-e d}\right )}{4 \sqrt {-e d}}\)	\(83\)
default	\(\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}\)	\(232\)

[In]

int((e*x^2+d)/(e^2*x^4+d^2),x,method=_RETURNVERBOSE)

[Out]

-1/4*2^(1/2)/(-e*d)^(1/2)*ln(e*x^2*(-e*d)^(1/2)-d*e*x*2^(1/2)-d*(-e*d)^(1/2))+1/4*2^(1/2)/(-e*d)^(1/2)*ln(e*x^
2*(-e*d)^(1/2)+d*e*x*2^(1/2)-d*(-e*d)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.83 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {2} \sqrt {-d e} \log \left (\frac {e^{2} x^{4} - 4 \, d e x^{2} - 2 \, \sqrt {2} {\left (e x^{3} - d x\right )} \sqrt {-d e} + d^{2}}{e^{2} x^{4} + d^{2}}\right )}{4 \, d e}, \frac {\sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} \sqrt {d e} x}{2 \, d}\right ) + \sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} {\left (e x^{3} + d x\right )} \sqrt {d e}}{2 \, d^{2}}\right )}{2 \, d e}\right ] \]

[In]

integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="fricas")

[Out]

[-1/4*sqrt(2)*sqrt(-d*e)*log((e^2*x^4 - 4*d*e*x^2 - 2*sqrt(2)*(e*x^3 - d*x)*sqrt(-d*e) + d^2)/(e^2*x^4 + d^2))
/(d*e), 1/2*(sqrt(2)*sqrt(d*e)*arctan(1/2*sqrt(2)*sqrt(d*e)*x/d) + sqrt(2)*sqrt(d*e)*arctan(1/2*sqrt(2)*(e*x^3
+ d*x)*sqrt(d*e)/d^2))/(d*e)]

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=- \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (- \sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (\sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} \]

[In]

integrate((e*x**2+d)/(e**2*x**4+d**2),x)

[Out]

-sqrt(2)*sqrt(-1/(d*e))*log(-sqrt(2)*d*x*sqrt(-1/(d*e)) - d/e + x**2)/4 + sqrt(2)*sqrt(-1/(d*e))*log(sqrt(2)*d
*x*sqrt(-1/(d*e)) - d/e + x**2)/4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (51) = 102\).

Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.03 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2} {\left (e + \sqrt {e^{2}}\right )} \log \left (\frac {2 \, \sqrt {e^{2}} x + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} - \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}{2 \, \sqrt {e^{2}} x + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}\right )}{8 \, \sqrt {e^{2}} \sqrt {-d \sqrt {e^{2}}}} + \frac {\sqrt {2} {\left (e + \sqrt {e^{2}}\right )} \log \left (\frac {2 \, \sqrt {e^{2}} x - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} - \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}{2 \, \sqrt {e^{2}} x - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}\right )}{8 \, \sqrt {e^{2}} \sqrt {-d \sqrt {e^{2}}}} - \frac {\sqrt {2} {\left (e - \sqrt {e^{2}}\right )} \log \left (\sqrt {e^{2}} x^{2} + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} x + d\right )}{8 \, \sqrt {d} {\left (e^{2}\right )}^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (e - \sqrt {e^{2}}\right )} \log \left (\sqrt {e^{2}} x^{2} - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} x + d\right )}{8 \, \sqrt {d} {\left (e^{2}\right )}^{\frac {3}{4}}} \]

[In]

integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*(e + sqrt(e^2))*log((2*sqrt(e^2)*x + sqrt(2)*sqrt(d)*(e^2)^(1/4) - sqrt(2)*sqrt(-d*sqrt(e^2)))/(2*
sqrt(e^2)*x + sqrt(2)*sqrt(d)*(e^2)^(1/4) + sqrt(2)*sqrt(-d*sqrt(e^2))))/(sqrt(e^2)*sqrt(-d*sqrt(e^2))) + 1/8*
sqrt(2)*(e + sqrt(e^2))*log((2*sqrt(e^2)*x - sqrt(2)*sqrt(d)*(e^2)^(1/4) - sqrt(2)*sqrt(-d*sqrt(e^2)))/(2*sqrt
(e^2)*x - sqrt(2)*sqrt(d)*(e^2)^(1/4) + sqrt(2)*sqrt(-d*sqrt(e^2))))/(sqrt(e^2)*sqrt(-d*sqrt(e^2))) - 1/8*sqrt
(2)*(e - sqrt(e^2))*log(sqrt(e^2)*x^2 + sqrt(2)*sqrt(d)*(e^2)^(1/4)*x + d)/(sqrt(d)*(e^2)^(3/4)) + 1/8*sqrt(2)
*(e - sqrt(e^2))*log(sqrt(e^2)*x^2 - sqrt(2)*sqrt(d)*(e^2)^(1/4)*x + d)/(sqrt(d)*(e^2)^(3/4))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2} \sqrt {-d e} \log \left (x^{2} + \sqrt {2} x \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{e^{2}}}\right )}{4 \, d e} - \frac {\sqrt {2} \sqrt {-d e} \log \left (x^{2} - \sqrt {2} x \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{e^{2}}}\right )}{4 \, d e} \]

[In]

integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*sqrt(-d*e)*log(x^2 + sqrt(2)*x*(d^2/e^2)^(1/4) + sqrt(d^2/e^2))/(d*e) - 1/4*sqrt(2)*sqrt(-d*e)*log
(x^2 - sqrt(2)*x*(d^2/e^2)^(1/4) + sqrt(d^2/e^2))/(d*e)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,e^{3/2}\,x^3}{2\,d^{3/2}}+\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )\right )}{4\,\sqrt {d}\,\sqrt {e}} \]

[In]

int((d + e*x^2)/(d^2 + e^2*x^4),x)

[Out]

(2^(1/2)*(2*atan((2^(1/2)*e^(1/2)*x)/(2*d^(1/2))) + 2*atan((2^(1/2)*e^(3/2)*x^3)/(2*d^(3/2)) + (2^(1/2)*e^(1/2
)*x)/(2*d^(1/2)))))/(4*d^(1/2)*e^(1/2))