3.89 \(\int \frac{1}{a^2+x^2} \, dx\)

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

[Out]

ArcTan[x/a]/a

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Rubi [A]  time = 0.00697499, 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 \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + x^2)^(-1),x]

[Out]

ArcTan[x/a]/a

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Rubi in Sympy [A]  time = 0.730526, size = 5, normalized size = 0.5 \[ \frac{\operatorname{atan}{\left (\frac{x}{a} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a**2+x**2),x)

[Out]

atan(x/a)/a

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Mathematica [A]  time = 0.00315407, size = 10, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + x^2)^(-1),x]

[Out]

ArcTan[x/a]/a

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Maple [A]  time = 0.013, size = 11, normalized size = 1.1 \[{\frac{1}{a}\arctan \left ({\frac{x}{a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a^2+x^2),x)

[Out]

arctan(x/a)/a

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Maxima [A]  time = 1.57168, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/a)/a

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Fricas [A]  time = 0.19404, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/a)/a

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Sympy [A]  time = 0.11201, size = 20, normalized size = 2. \[ \frac{- \frac{i \log{\left (- i a + x \right )}}{2} + \frac{i \log{\left (i a + x \right )}}{2}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a**2+x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22019, size = 14, normalized size = 1.4 \[ \frac{\arctan \left (\frac{x}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x/a)/a