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

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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {b x}{a}\right )}{a b} \]

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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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {b x}{a}\right )}{a b} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + b^2*x^2)^(-1),x]

[Out]

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

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fricas [A]  time = 0.59, size = 14, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

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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giac [A]  time = 0.92, size = 14, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

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)

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maple [A]  time = 0.31, size = 15, normalized size = 1.07

method	result	size
default	\(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\)	\(15\)
risch	\(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 14, normalized size = 1.00 \[ \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

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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mupad [B]  time = 0.04, size = 14, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (\frac {b\,x}{a}\right )}{a\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + b^2*x^2),x)

[Out]

atan((b*x)/a)/(a*b)

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sympy [C]  time = 0.14, size = 26, normalized size = 1.86 \[ \frac {- \frac {i \log {\left (- \frac {i a}{b} + x \right )}}{2} + \frac {i \log {\left (\frac {i a}{b} + x \right )}}{2}}{a b} \]

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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