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 = 0.00771223, 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 \[ \frac{\tanh ^{-1}\left (\frac{x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]  Int[(c^2 - x^2)^(-1),x]

[Out]

ArcTanh[x/c]/c

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Rubi in Sympy [A]  time = 0.898045, size = 5, normalized size = 0.5 \[ \frac{\operatorname{atanh}{\left (\frac{x}{c} \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c**2-x**2),x)

[Out]

atanh(x/c)/c

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Mathematica [A]  time = 0.00355437, 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 = 0.003, size = 22, normalized size = 2.2 \[ -{\frac{\ln \left ( -c+x \right ) }{2\,c}}+{\frac{\ln \left ( c+x \right ) }{2\,c}} \]

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(c+x)

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Maxima [A]  time = 1.40752, size = 28, normalized size = 2.8 \[ \frac{\log \left (c + x\right )}{2 \, c} - \frac{\log \left (-c + x\right )}{2 \, c} \]

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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Fricas [A]  time = 0.224707, size = 24, normalized size = 2.4 \[ \frac{\log \left (c + x\right ) - \log \left (-c + x\right )}{2 \, c} \]

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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Sympy [A]  time = 0.114832, size = 15, normalized size = 1.5 \[ - \frac{\frac{\log{\left (- c + x \right )}}{2} - \frac{\log{\left (c + x \right )}}{2}}{c} \]

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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GIAC/XCAS [A]  time = 0.197995, size = 31, normalized size = 3.1 \[ \frac{{\rm ln}\left ({\left | c + x \right |}\right )}{2 \, c} - \frac{{\rm ln}\left ({\left | -c + x \right |}\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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