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

Optimal. Leaf size=64 \[ -\frac{1}{4 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0643417, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4934, 4884} \[ -\frac{1}{4 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4934

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q
+ 1))/(4*c^3*d*(q + 1)^2), x] + (-Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x],
x] + Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]

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{x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{1}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{2 a^2 c}\\ &=-\frac{1}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0429549, size = 47, normalized size = 0.73 \[ \frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-2 a x \tan ^{-1}(a x)-1}{4 a^3 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*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*a^3*c^2*(1 + a^2*x^2))

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Maple [A]  time = 0.036, size = 59, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{3}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{x\arctan \left ( ax \right ) }{2\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{a}^{3}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.61264, size = 112, normalized size = 1.75 \begin{align*} -\frac{1}{2} \,{\left (\frac{x}{a^{4} c^{2} x^{2} + a^{2} c^{2}} - \frac{\arctan \left (a x\right )}{a^{3} 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^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x/(a^4*c^2*x^2 + a^2*c^2) - arctan(a*x)/(a^3*c^2))*arctan(a*x) - 1/4*((a^2*x^2 + 1)*arctan(a*x)^2 + 1)*a
/(a^6*c^2*x^2 + a^4*c^2)

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Fricas [A]  time = 1.61264, size = 113, normalized size = 1.77 \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^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*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^5*c^2*x^2 + a^3*c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{2} \operatorname{atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*atan(a*x)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \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(x^2*arctan(a*x)/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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