3.554 \(\int \frac{1}{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\frac{e x}{d}\right )}{d e} \]

[Out]

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

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Rubi [A]  time = 0.0169764, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\tanh ^{-1}\left (\frac{e x}{d}\right )}{d e} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.51866, size = 8, normalized size = 0.57 \[ \frac{\operatorname{atanh}{\left (\frac{e x}{d} \right )}}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-e**2*x**2+d**2),x)

[Out]

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

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Mathematica [A]  time = 0.00439945, size = 14, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{e x}{d}\right )}{d e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(-1),x]

[Out]

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

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Maple [B]  time = 0.007, size = 32, normalized size = 2.3 \[ -{\frac{\ln \left ( ex-d \right ) }{2\,ed}}+{\frac{\ln \left ( ex+d \right ) }{2\,ed}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-e^2*x^2+d^2),x)

[Out]

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

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Maxima [A]  time = 0.785331, size = 42, normalized size = 3. \[ \frac{\log \left (e x + d\right )}{2 \, d e} - \frac{\log \left (e x - d\right )}{2 \, d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(e^2*x^2 - d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248668, size = 34, normalized size = 2.43 \[ \frac{\log \left (e x + d\right ) - \log \left (e x - d\right )}{2 \, d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(e^2*x^2 - d^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.150357, size = 20, normalized size = 1.43 \[ - \frac{\frac{\log{\left (- \frac{d}{e} + x \right )}}{2} - \frac{\log{\left (\frac{d}{e} + x \right )}}{2}}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-e**2*x**2+d**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229488, size = 51, normalized size = 3.64 \[ -\frac{e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(e^2*x^2 - d^2),x, algorithm="giac")

[Out]

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