3.31 \(\int \frac{1}{c^2+x^2} \, dx\)

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

[Out]

ArcTan[x/c]/c

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Rubi [A]  time = 0.00715066, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\frac{x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]  Int[(c^2 + x^2)^(-1),x]

[Out]

ArcTan[x/c]/c

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Rubi in Sympy [A]  time = 0.726118, size = 5, normalized size = 0.5 \[ \frac{\operatorname{atan}{\left (\frac{x}{c} \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c**2+x**2),x)

[Out]

atan(x/c)/c

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Mathematica [A]  time = 0.00314959, size = 10, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(c^2 + x^2)^(-1),x]

[Out]

ArcTan[x/c]/c

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Maple [A]  time = 0.003, size = 11, normalized size = 1.1 \[{\frac{1}{c}\arctan \left ({\frac{x}{c}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c^2+x^2),x)

[Out]

arctan(x/c)/c

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Maxima [A]  time = 1.51966, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{c}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/c)/c

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Fricas [A]  time = 0.231272, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{c}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/c)/c

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Sympy [A]  time = 0.102591, size = 20, normalized size = 2. \[ \frac{- \frac{i \log{\left (- i c + x \right )}}{2} + \frac{i \log{\left (i c + x \right )}}{2}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c**2+x**2),x)

[Out]

(-I*log(-I*c + x)/2 + I*log(I*c + x)/2)/c

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GIAC/XCAS [A]  time = 0.199612, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{c}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/c)/c