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 = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(c^2 - x^2)^(-1),x]

[Out]

ArcTanh[x/c]/c

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fricas [A]  time = 0.40, size = 18, normalized size = 1.80 \[ \frac {\log \left (c + x\right ) - \log \left (-c + x\right )}{2 \, c} \]

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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giac [B]  time = 0.95, size = 23, normalized size = 2.30 \[ \frac {\log \left ({\left | c + x \right |}\right )}{2 \, c} - \frac {\log \left ({\left | -c + x \right |}\right )}{2 \, c} \]

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

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maple [B]  time = 0.01, size = 22, normalized size = 2.20 \[ -\frac {\ln \left (-c +x \right )}{2 c}+\frac {\ln \left (c +x \right )}{2 c} \]

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(x+c)

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maxima [B]  time = 0.43, size = 21, normalized size = 2.10 \[ \frac {\log \left (c + x\right )}{2 \, c} - \frac {\log \left (-c + x\right )}{2 \, c} \]

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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mupad [B]  time = 0.16, size = 10, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\frac {x}{c}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c^2 - x^2),x)

[Out]

atanh(x/c)/c

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sympy [B]  time = 0.12, size = 15, normalized size = 1.50 \[ - \frac {\frac {\log {\left (- c + x \right )}}{2} - \frac {\log {\left (c + x \right )}}{2}}{c} \]

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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