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

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]

[Out]

ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e])

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Rubi [A]  time = 0.0227719, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e])

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Rubi in Sympy [A]  time = 5.60839, size = 22, normalized size = 0.92 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{\sqrt{d} \sqrt{e}} \]

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]

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

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Mathematica [A]  time = 0.00653757, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e])

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Maple [A]  time = 0.002, size = 16, normalized size = 0.7 \[{1{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

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

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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

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Fricas [A]  time = 0.271492, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{2 \, d e x +{\left (e x^{2} + d\right )} \sqrt{d e}}{e x^{2} - d}\right )}{2 \, \sqrt{d e}}, \frac{\arctan \left (\frac{\sqrt{-d e} x}{d}\right )}{\sqrt{-d e}}\right ] \]

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]

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

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Sympy [A]  time = 0.33632, size = 46, normalized size = 1.92 \[ - \frac{\sqrt{\frac{1}{d e}} \log{\left (- d \sqrt{\frac{1}{d e}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{d e}} \log{\left (d \sqrt{\frac{1}{d e}} + x \right )}}{2} \]

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

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GIAC/XCAS [A]  time = 0.27893, size = 157, normalized size = 6.54 \[ \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-4\right )}}{2 \, d^{2}} + \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} - \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} \]

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]

1/2*((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^2 + 1/4*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*e^(-6)*l
n(abs((d^2)^(1/4)*e^(-1/2) + x))/d^2 - 1/4*((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^2