3.32 \(\int \frac{1}{c^2-x^2} \, dx\)

Optimal. Leaf size=10 \[ \frac{\tanh ^{-1}\left (\frac{x}{c}\right )}{c} \]

[Out]

ArcTanh[x/c]/c

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Rubi [A]  time = 0.0028164, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {206} \[ \frac{\tanh ^{-1}\left (\frac{x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[(c^2 - x^2)^(-1),x]

[Out]

ArcTanh[x/c]/c

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0025066, size = 10, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(c^2 - x^2)^(-1),x]

[Out]

ArcTanh[x/c]/c

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Maple [B]  time = 0.004, size = 22, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ( -c+x \right ) }{2\,c}}+{\frac{\ln \left ( c+x \right ) }{2\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c^2-x^2),x)

[Out]

-1/2/c*ln(-c+x)+1/2/c*ln(c+x)

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Maxima [A]  time = 0.946434, size = 28, normalized size = 2.8 \begin{align*} \frac{\log \left (c + x\right )}{2 \, c} - \frac{\log \left (-c + x\right )}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c^2-x^2),x, algorithm="maxima")

[Out]

1/2*log(c + x)/c - 1/2*log(-c + x)/c

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Fricas [A]  time = 1.90468, size = 46, normalized size = 4.6 \begin{align*} \frac{\log \left (c + x\right ) - \log \left (-c + x\right )}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c^2-x^2),x, algorithm="fricas")

[Out]

1/2*(log(c + x) - log(-c + x))/c

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Sympy [B]  time = 0.112406, size = 15, normalized size = 1.5 \begin{align*} - \frac{\frac{\log{\left (- c + x \right )}}{2} - \frac{\log{\left (c + x \right )}}{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c**2-x**2),x)

[Out]

-(log(-c + x)/2 - log(c + x)/2)/c

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Giac [B]  time = 1.07247, size = 31, normalized size = 3.1 \begin{align*} \frac{\log \left ({\left | c + x \right |}\right )}{2 \, c} - \frac{\log \left ({\left | -c + x \right |}\right )}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c^2-x^2),x, algorithm="giac")

[Out]

1/2*log(abs(c + x))/c - 1/2*log(abs(-c + x))/c