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

Optimal result

Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {211} \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[In]

Int[(a^2 + b^2*x^2)^(-1),x]

[Out]

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

Rule 211

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

Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {b x}{a}\right )}{a b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[In]

Integrate[(a^2 + b^2*x^2)^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\)	\(15\)
risch	\(\frac {\arctan \left (\frac {b x}{a}\right )}{a b}\)	\(15\)
parallelrisch	\(-\frac {i \ln \left (-i a +b x \right )-i \ln \left (i a +b x \right )}{2 a b}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {- \frac {i \log {\left (- \frac {i a}{b} + x \right )}}{2} + \frac {i \log {\left (\frac {i a}{b} + x \right )}}{2}}{a b} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\arctan \left (\frac {b x}{a}\right )}{a b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 x^2} \, dx=\frac {\mathrm {atan}\left (\frac {b\,x}{a}\right )}{a\,b} \]

[In]

int(1/(a^2 + b^2*x^2),x)

[Out]

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