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 = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \]

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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fricas [A]  time = 0.38, size = 25, normalized size = 1.79 \[ \frac {\log \left (e x + d\right ) - \log \left (e x - d\right )}{2 \, d e} \]

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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giac [B]  time = 0.46, size = 38, normalized size = 2.71 \[ -\frac {e^{\left (-1\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} \]

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)

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maple [B]  time = 0.01, size = 32, normalized size = 2.29 \[ -\frac {\ln \left (e x -d \right )}{2 d e}+\frac {\ln \left (e x +d \right )}{2 d e} \]

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 [B]  time = 0.96, size = 31, normalized size = 2.21 \[ \frac {\log \left (e x + d\right )}{2 \, d e} - \frac {\log \left (e x - d\right )}{2 \, d e} \]

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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mupad [B]  time = 3.43, size = 14, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )}{d\,e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d^2 - e^2*x^2),x)

[Out]

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

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sympy [B]  time = 0.14, 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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