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

Optimal. Leaf size=14 \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

[Out]

ArcTan[(b*x)/a]/(a*b)

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Rubi [A]  time = 0.0048183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + b^2*x^2)^(-1),x]

[Out]

ArcTan[(b*x)/a]/(a*b)

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.0030547, size = 14, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + b^2*x^2)^(-1),x]

[Out]

ArcTan[(b*x)/a]/(a*b)

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Maple [A]  time = 0.003, size = 15, normalized size = 1.1 \begin{align*}{\frac{1}{ab}\arctan \left ({\frac{bx}{a}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^2+a^2),x)

[Out]

arctan(b*x/a)/a/b

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Maxima [A]  time = 1.40985, size = 19, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+a^2),x, algorithm="maxima")

[Out]

arctan(b*x/a)/(a*b)

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Fricas [A]  time = 1.95438, size = 28, normalized size = 2. \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+a^2),x, algorithm="fricas")

[Out]

arctan(b*x/a)/(a*b)

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Sympy [C]  time = 0.125087, size = 26, normalized size = 1.86 \begin{align*} \frac{- \frac{i \log{\left (- \frac{i a}{b} + x \right )}}{2} + \frac{i \log{\left (\frac{i a}{b} + x \right )}}{2}}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**2+a**2),x)

[Out]

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

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Giac [A]  time = 1.0527, size = 19, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^2+a^2),x, algorithm="giac")

[Out]

arctan(b*x/a)/(a*b)