3.195 \(\int \frac{\left (d+e x^2\right )^2}{d^2-e^2 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0407428, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)^2/(d^2 - e^2*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.7222, size = 26, normalized size = 0.9 \[ \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{\sqrt{e}} - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d)*atanh(sqrt(e)*x/sqrt(d))/sqrt(e) - x

_______________________________________________________________________________________

Mathematica [A]  time = 0.0138527, size = 29, normalized size = 1. \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.8 \[ -x+2\,{\frac{d}{\sqrt{de}}{\it Artanh} \left ({\frac{ex}{\sqrt{de}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^2/(-e^2*x^4+d^2),x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.277895, size = 1, normalized size = 0.03 \[ \left [\sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - x, 2 \, \sqrt{-\frac{d}{e}} \arctan \left (\frac{x}{\sqrt{-\frac{d}{e}}}\right ) - x\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[sqrt(d/e)*log((e*x^2 + 2*e*x*sqrt(d/e) + d)/(e*x^2 - d)) - x, 2*sqrt(-d/e)*arct
an(x/sqrt(-d/e)) - x]

_______________________________________________________________________________________

Sympy [A]  time = 1.31547, size = 34, normalized size = 1.17 \[ - x - \sqrt{\frac{d}{e}} \log{\left (x - \sqrt{\frac{d}{e}} \right )} + \sqrt{\frac{d}{e}} \log{\left (x + \sqrt{\frac{d}{e}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - sqrt(d/e)*log(x - sqrt(d/e)) + sqrt(d/e)*log(x + sqrt(d/e))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.280529, size = 159, normalized size = 5.48 \[ \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-4\right )}}{d} + \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{2 \, d} - \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{2 \, d} - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((d^2)^(1/4)*d*e^(7/2) - (d^2)^(1/4)*abs(d)*e^(7/2))*arctan(x*e^(1/2)/(d^2)^(1/4
))*e^(-4)/d + 1/2*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*e^(-6)*ln(abs(
(d^2)^(1/4)*e^(-1/2) + x))/d - 1/2*((d^2)^(1/4)*d*e^(7/2) + (d^2)^(1/4)*abs(d)*e
^(7/2))*e^(-4)*ln(abs(-(d^2)^(1/4)*e^(-1/2) + x))/d - x