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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = 0.05, size = 17, normalized size = 1.00 \[ \frac {\log \left (2 \cot ^{-1}(x)-1\right )}{2 a b} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 15, normalized size = 0.88 \[ \frac {\log \left (2 \, \operatorname {arccot}\relax (x) - 1\right )}{2 \, a b} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.11, size = 18, normalized size = 1.06 \[ \frac {\log \left ({\left | 2 \, \arctan \left (\frac {1}{x}\right ) - 1 \right |}\right )}{2 \, a b} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 19, normalized size = 1.12 \[ \frac {\ln \left (2 b \,\mathrm {arccot}\relax (x )-b \right )}{2 a b} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 17, normalized size = 1.00 \[ \frac {\log \left ({\left | 2 \, \arctan \left (1, x\right ) - 1 \right |}\right )}{2 \, a b} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 15, normalized size = 0.88 \[ \frac {\ln \left (2\,\mathrm {acot}\relax (x)-1\right )}{2\,a\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*acot(x) - 1)/(2*a*b)

________________________________________________________________________________________

sympy [A]  time = 0.53, size = 12, normalized size = 0.71 \[ \frac {\log {\left (\operatorname {acot}{\relax (x )} - \frac {1}{2} \right )}}{2 a b} \]

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)

________________________________________________________________________________________