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

Optimal. Leaf size=17 \[ -\frac{\log \left (1-2 \tan ^{-1}(x)\right )}{2 a b} \]

[Out]

-Log[1 - 2*ArcTan[x]]/(2*a*b)

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Rubi [A]  time = 0.0393312, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4882} \[ -\frac{\log \left (1-2 \tan ^{-1}(x)\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*x^2)*(b - 2*b*ArcTan[x])),x]

[Out]

-Log[1 - 2*ArcTan[x]]/(2*a*b)

Rule 4882

Int[1/(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[Log[RemoveContent[a + b*Ar
cTan[c*x], x]]/(b*c*d), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

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

Mathematica [A]  time = 0.0445816, size = 17, normalized size = 1. \[ -\frac{\log \left (2 \tan ^{-1}(x)-1\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*x^2)*(b - 2*b*ArcTan[x])),x]

[Out]

-Log[-1 + 2*ArcTan[x]]/(2*a*b)

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Maple [A]  time = 0.073, size = 19, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( 2\,b\arctan \left ( x \right ) -b \right ) }{2\,ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a*ln(2*b*arctan(x)-b)/b

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Maxima [A]  time = 0.969344, size = 22, normalized size = 1.29 \begin{align*} -\frac{\log \left ({\left | 2 \, \arctan \left (x\right ) - 1 \right |}\right )}{2 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(2*arctan(x) - 1))/(a*b)

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Fricas [A]  time = 1.79571, size = 45, normalized size = 2.65 \begin{align*} -\frac{\log \left (2 \, \arctan \left (x\right ) - 1\right )}{2 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(2*arctan(x) - 1)/(a*b)

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Sympy [A]  time = 0.70524, size = 14, normalized size = 0.82 \begin{align*} - \frac{\log{\left (\operatorname{atan}{\left (x \right )} - \frac{1}{2} \right )}}{2 a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(atan(x) - 1/2)/(2*a*b)

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Giac [A]  time = 1.09499, size = 22, normalized size = 1.29 \begin{align*} -\frac{\log \left ({\left | 2 \, \arctan \left (x\right ) - 1 \right |}\right )}{2 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(2*arctan(x) - 1))/(a*b)