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

Optimal. Leaf size=16 \[ \frac{\tan ^{-1}(a x)^2}{2 a c} \]

[Out]

ArcTan[a*x]^2/(2*a*c)

________________________________________________________________________________________

Rubi [A]  time = 0.0173921, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {4884} \[ \frac{\tan ^{-1}(a x)^2}{2 a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(c + a^2*c*x^2),x]

[Out]

ArcTan[a*x]^2/(2*a*c)

Rule 4884

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

Rubi steps

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

Mathematica [A]  time = 0.0027385, size = 16, normalized size = 1. \[ \frac{\tan ^{-1}(a x)^2}{2 a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[a*x]^2/(2*a*c)

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(a*x)^2/a/c

________________________________________________________________________________________

Maxima [A]  time = 1.54783, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{2}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(a*x)^2/(a*c)

________________________________________________________________________________________

Fricas [A]  time = 1.6202, size = 34, normalized size = 2.12 \begin{align*} \frac{\arctan \left (a x\right )^{2}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(a*x)^2/(a*c)

________________________________________________________________________________________

Sympy [A]  time = 2.4263, size = 36, normalized size = 2.25 \begin{align*} \begin{cases} 0 & \text{for}\: a = 0 \\\tilde{\infty } \left (\begin{cases} 0 & \text{for}\: a = 0 \\\frac{a x \operatorname{atan}{\left (a x \right )} - \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{2}}{a} & \text{otherwise} \end{cases}\right ) & \text{for}\: c = 0 \\\frac{\operatorname{atan}^{2}{\left (a x \right )}}{2 a c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((0, Eq(a, 0)), (zoo*Piecewise((0, Eq(a, 0)), ((a*x*atan(a*x) - log(a**2*x**2 + 1)/2)/a, True)), Eq(c
, 0)), (atan(a*x)**2/(2*a*c), True))

________________________________________________________________________________________

Giac [A]  time = 1.11326, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{2}}{2 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(a*x)^2/(a*c)