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 = 0.0078265, 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, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{e x}{d}\right )}{d e} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(-1),x]

[Out]

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{d^2-e^2 x^2} \, dx &=\frac{\tanh ^{-1}\left (\frac{e x}{d}\right )}{d e}\\ \end{align*}

Mathematica [A]  time = 0.0029067, 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 = 0.006, size = 32, normalized size = 2.3 \begin{align*}{\frac{\ln \left ( ex+d \right ) }{2\,de}}-{\frac{\ln \left ( ex-d \right ) }{2\,de}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.974911, size = 42, normalized size = 3. \begin{align*} \frac{\log \left (e x + d\right )}{2 \, d e} - \frac{\log \left (e x - d\right )}{2 \, d e} \end{align*}

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 = 1.20058, size = 55, normalized size = 3.93 \begin{align*} \frac{\log \left (e x + d\right ) - \log \left (e x - d\right )}{2 \, d e} \end{align*}

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 [B]  time = 0.133459, size = 20, normalized size = 1.43 \begin{align*} - \frac{\frac{\log{\left (- \frac{d}{e} + x \right )}}{2} - \frac{\log{\left (\frac{d}{e} + x \right )}}{2}}{d e} \end{align*}

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 [B]  time = 1.3071, size = 51, normalized size = 3.64 \begin{align*} -\frac{e^{\left (-1\right )} \log \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 |}} \end{align*}

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