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

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 = 0.0025235, 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, Rules used = {203} \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + x^2)^(-1),x]

[Out]

ArcTan[x/a]/a

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0022225, 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 = 0.003, size = 11, normalized size = 1.1 \begin{align*}{\frac{1}{a}\arctan \left ({\frac{x}{a}} \right ) } \end{align*}

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 = 1.68187, size = 14, normalized size = 1.4 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{a} \end{align*}

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 = 1.31858, size = 20, normalized size = 2. \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{a} \end{align*}

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 [C]  time = 0.101606, size = 20, normalized size = 2. \begin{align*} \frac{- \frac{i \log{\left (- i a + x \right )}}{2} + \frac{i \log{\left (i a + x \right )}}{2}}{a} \end{align*}

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 [A]  time = 1.2022, size = 14, normalized size = 1.4 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{a} \end{align*}

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