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

Optimal. Leaf size=75 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

[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])

_______________________________________________________________________________________

Rubi [A]  time = 0.0971491, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

[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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.5742, size = 73, normalized size = 0.97 \[ - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{d} \sqrt{e}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{d} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)/(e**2*x**4+d**2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0510194, size = 60, normalized size = 0.8 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )-\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

[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 [B]  time = 0.013, size = 290, normalized size = 3.9 \[{\frac{\sqrt{2}}{8\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ) }+{\frac{\sqrt{2}}{8\,e}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)/(e^2*x^4+d^2),x)

[Out]

1/8/d*(d^2/e^2)^(1/4)*2^(1/2)*ln((x^2+(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2))
/(x^2-(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2)))+1/4/d*(d^2/e^2)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(d^2/e^2)^(1/4)*x+1)+1/4/d*(d^2/e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(d^2/e^2)^(1/4)*x-1)+1/8/e/(d^2/e^2)^(1/4)*2^(1/2)*ln((x^2-(d^2/e^2)^(1/4)*x*2^
(1/2)+(d^2/e^2)^(1/2))/(x^2+(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2)))+1/4/e/(d
^2/e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(d^2/e^2)^(1/4)*x+1)+1/4/e/(d^2/e^2)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(d^2/e^2)^(1/4)*x-1)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.289028, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2} \log \left (\frac{4 \, d e^{2} x^{3} - 4 \, d^{2} e x + \sqrt{2}{\left (e^{2} x^{4} - 4 \, d e x^{2} + d^{2}\right )} \sqrt{-d e}}{e^{2} x^{4} + d^{2}}\right )}{4 \, \sqrt{-d e}}, \frac{\sqrt{2}{\left (\arctan \left (\frac{\sqrt{2} \sqrt{d e} x}{2 \, d}\right ) + \arctan \left (\frac{\sqrt{2}{\left (e^{2} x^{3} + d e x\right )}}{2 \, \sqrt{d e} d}\right )\right )}}{2 \, \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.464425, size = 87, normalized size = 1.16 \[ - \frac{\sqrt{2} \sqrt{- \frac{1}{d e}} \log{\left (- \sqrt{2} d x \sqrt{- \frac{1}{d e}} - \frac{d}{e} + x^{2} \right )}}{4} + \frac{\sqrt{2} \sqrt{- \frac{1}{d e}} \log{\left (\sqrt{2} d x \sqrt{- \frac{1}{d e}} - \frac{d}{e} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27754, size = 300, normalized size = 4. \[ \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} - 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} - \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (-\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*arctan(1/2*sqrt(2)*(
sqrt(2)*(d^2)^(1/4)*e^(-1/2) + 2*x)*e^(1/2)/(d^2)^(1/4))*e^(-6)/d^2 + 1/4*sqrt(2
)*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*(
d^2)^(1/4)*e^(-1/2) - 2*x)*e^(1/2)/(d^2)^(1/4))*e^(-6)/d^2 + 1/8*sqrt(2)*((d^2)^
(1/4)*d*e^(11/2) - (d^2)^(3/4)*e^(11/2))*e^(-6)*ln(sqrt(2)*(d^2)^(1/4)*x*e^(-1/2
) + x^2 + sqrt(d^2)*e^(-1))/d^2 - 1/8*sqrt(2)*((d^2)^(1/4)*d*e^(11/2) - (d^2)^(3
/4)*e^(11/2))*e^(-6)*ln(-sqrt(2)*(d^2)^(1/4)*x*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1)
)/d^2