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

Optimal. Leaf size=12 \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]

[Out]

Log[ArcTan[a*x]]/(a*c)

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Rubi [A]  time = 0.0264302, antiderivative size = 12, 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} \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcTan[a*x]]/(a*c)

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

Mathematica [A]  time = 0.0138428, size = 12, normalized size = 1. \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcTan[a*x]]/(a*c)

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Maple [A]  time = 0.05, size = 13, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \arctan \left ( ax \right ) \right ) }{ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(arctan(a*x))/a/c

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Maxima [A]  time = 1.05453, size = 20, normalized size = 1.67 \begin{align*} \frac{\log \left (2 \,{\left | \arctan \left (a x\right ) \right |}\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*abs(arctan(a*x)))/(a*c)

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Fricas [A]  time = 1.85804, size = 32, normalized size = 2.67 \begin{align*} \frac{\log \left (\arctan \left (a x\right )\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arctan(a*x))/(a*c)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.15777, size = 18, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | \arctan \left (a x\right ) \right |}\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(arctan(a*x)))/(a*c)