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

Optimal. Leaf size=106 \[ \frac{x}{4 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\tan ^{-1}(a x)}{4 a^3 c^2} \]

[Out]

x/(4*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^3*c^2) - ArcTan[a*x]/(2*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]
^2)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a^3*c^2)

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Rubi [A]  time = 0.110012, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4936, 4930, 199, 205} \[ \frac{x}{4 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\tan ^{-1}(a x)}{4 a^3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^3*c^2) - ArcTan[a*x]/(2*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]
^2)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a^3*c^2)

Rule 4936

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(a + b*ArcTan
[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (Dist[(b*p)/(2*c), Int[(x*(a + b*ArcTan[c*x])^(p - 1))/(d + e*x^2)^
2, x], x] - Simp[(x*(a + b*ArcTan[c*x])^p)/(2*c^2*d*(d + e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c
^2*d] && GtQ[p, 0]

Rule 4930

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a}\\ &=-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2}\\ &=\frac{x}{4 a^2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\int \frac{1}{c+a^2 c x^2} \, dx}{4 a^2 c}\\ &=\frac{x}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)}{4 a^3 c^2}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0904656, size = 68, normalized size = 0.64 \[ \frac{2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3+3 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+3 a x-6 a x \tan ^{-1}(a x)^2}{12 a^3 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x + 3*(-1 + a^2*x^2)*ArcTan[a*x] - 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x^2)*ArcTan[a*x]^3)/(12*a^3*c^2*(1 +
a^2*x^2))

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Maple [A]  time = 0.035, size = 97, normalized size = 0.9 \begin{align*}{\frac{x}{4\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{\arctan \left ( ax \right ) }{4\,{a}^{3}{c}^{2}}}-{\frac{\arctan \left ( ax \right ) }{2\,{a}^{3}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{6\,{a}^{3}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.63624, size = 204, normalized size = 1.92 \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 )^{2} + \frac{{\left (2 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 3 \, a x + 3 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{12 \,{\left (a^{7} c^{2} x^{2} + a^{5} c^{2}\right )}} - \frac{{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a \arctan \left (a x\right )}{2 \,{\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)^2/(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)^2 + 1/12*(2*(a^2*x^2 + 1)*arctan(a*x)^3 +
3*a*x + 3*(a^2*x^2 + 1)*arctan(a*x))*a^2/(a^7*c^2*x^2 + a^5*c^2) - 1/2*((a^2*x^2 + 1)*arctan(a*x)^2 + 1)*a*ar
ctan(a*x)/(a^6*c^2*x^2 + a^4*c^2)

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Fricas [A]  time = 2.12255, size = 166, normalized size = 1.57 \begin{align*} -\frac{6 \, a x \arctan \left (a x\right )^{2} - 2 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \,{\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{12 \,{\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)^2/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/12*(6*a*x*arctan(a*x)^2 - 2*(a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x - 3*(a^2*x^2 - 1)*arctan(a*x))/(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}^{2}{\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)**2/(a**2*c*x**2+c)**2,x)

[Out]

Integral(x**2*atan(a*x)**2/(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 )^{2}}{{\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)^2/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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