3.6.54 \(\int \frac {1}{d^2-e^2 x^2} \, dx\) [554]

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 = 0.00, 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 = {214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(-1),x]

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], 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 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(30\) vs. \(2(14)=28\).
time = 0.07, size = 31, normalized size = 2.21

method	result	size
default	\(\frac {\ln \left (e x +d \right )}{2 d e}-\frac {\ln \left (-e x +d \right )}{2 d e}\)	\(31\)
norman	\(\frac {\ln \left (e x +d \right )}{2 d e}-\frac {\ln \left (-e x +d \right )}{2 d e}\)	\(31\)
risch	\(\frac {\ln \left (e x +d \right )}{2 d e}-\frac {\ln \left (-e x +d \right )}{2 d e}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
time = 0.30, size = 31, normalized size = 2.21 \begin {gather*} \frac {\log \left (e x + d\right )}{2 \, d e} - \frac {\log \left (e x - d\right )}{2 \, d e} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (14) = 28\).
time = 0.41, size = 39, normalized size = 2.79 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (\frac {x^{2} e^{2} + 2 \, d x e + d^{2}}{x^{2} e^{2} - d^{2}}\right )}{2 \, d} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
time = 0.05, size = 20, normalized size = 1.43 \begin {gather*} - \frac {\frac {\log {\left (- \frac {d}{e} + x \right )}}{2} - \frac {\log {\left (\frac {d}{e} + x \right )}}{2}}{d e} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
time = 4.07, size = 38, normalized size = 2.71 \begin {gather*} -\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 {gather*}

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

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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