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

Optimal. Leaf size=61 \[ \frac{1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2} \]

[Out]

1/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a*c^2)

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Rubi [A]  time = 0.0258874, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4892, 261} \[ \frac{1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(c + a^2*c*x^2)^2,x]

[Out]

1/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a*c^2)

Rule 4892

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTan[c*x])
^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + Simp
[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p,
0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac{x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}-\frac{1}{2} a \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac{1}{4 a c^2 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}\\ \end{align*}

Mathematica [A]  time = 0.0215124, size = 44, normalized size = 0.72 \[ \frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+1}{4 c^2 \left (a^3 x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + 2*a*x*ArcTan[a*x] + (1 + a^2*x^2)*ArcTan[a*x]^2)/(4*c^2*(a + a^3*x^2))

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Maple [A]  time = 0.034, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{4\,a{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x\arctan \left ( ax \right ) }{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,a{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a/c^2/(a^2*x^2+1)+1/2*x*arctan(a*x)/c^2/(a^2*x^2+1)+1/4*arctan(a*x)^2/a/c^2

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Maxima [A]  time = 1.66636, size = 105, normalized size = 1.72 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{a^{2} c^{2} x^{2} + c^{2}} + \frac{\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right ) - \frac{{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a}{4 \,{\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x/(a^2*c^2*x^2 + c^2) + arctan(a*x)/(a*c^2))*arctan(a*x) - 1/4*((a^2*x^2 + 1)*arctan(a*x)^2 - 1)*a/(a^4*c
^2*x^2 + a^2*c^2)

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Fricas [A]  time = 1.65389, size = 109, normalized size = 1.79 \begin{align*} \frac{2 \, a x \arctan \left (a x\right ) +{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1}{4 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*x*arctan(a*x) + (a^2*x^2 + 1)*arctan(a*x)^2 + 1)/(a^3*c^2*x^2 + a*c^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RecursionError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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