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

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

[Out]

Log[1 - 2*ArcCot[x]]/(2*a*b)

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Rubi [A]  time = 0.04, 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 = {4883} \[ \frac{\log \left (1-2 \cot ^{-1}(x)\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 - 2*ArcCot[x]]/(2*a*b)

Rule 4883

Int[1/(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[Log[RemoveContent[a + b*A
rcCot[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 \cot ^{-1}(x)\right )} \, dx &=\frac{\log \left (1-2 \cot ^{-1}(x)\right )}{2 a b}\\ \end{align*}

Mathematica [A]  time = 0.0447506, size = 17, normalized size = 1. \[ \frac{\log \left (2 \cot ^{-1}(x)-1\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-1 + 2*ArcCot[x]]/(2*a*b)

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Maple [A]  time = 0.128, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( 2\,b{\rm arccot} \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*arccot(x)),x)

[Out]

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

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Maxima [A]  time = 1.01412, size = 23, normalized size = 1.35 \begin{align*} \frac{\log \left ({\left | 2 \, \arctan \left (1, 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*arccot(x)),x, algorithm="maxima")

[Out]

1/2*log(abs(2*arctan2(1, x) - 1))/(a*b)

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Fricas [A]  time = 2.34995, size = 43, normalized size = 2.53 \begin{align*} \frac{\log \left (2 \, \operatorname{arccot}\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*arccot(x)),x, algorithm="fricas")

[Out]

1/2*log(2*arccot(x) - 1)/(a*b)

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Sympy [A]  time = 0.74049, size = 12, normalized size = 0.71 \begin{align*} \frac{\log{\left (\operatorname{acot}{\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*acot(x)),x)

[Out]

log(acot(x) - 1/2)/(2*a*b)

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Giac [A]  time = 1.14587, size = 24, normalized size = 1.41 \begin{align*} \frac{\log \left ({\left | 2 \, \arctan \left (\frac{1}{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*arccot(x)),x, algorithm="giac")

[Out]

1/2*log(abs(2*arctan(1/x) - 1))/(a*b)