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

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

[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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Rubi [A]  time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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]

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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.52, size = 49, normalized size = 0.77 \[ -\frac {2 \, a x \arctan \left (a x\right ) - {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1}{4 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.04, size = 59, normalized size = 0.92 \[ -\frac {1}{4 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {x \arctan \left (a x \right )}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{2}}{4 a^{3} c^{2}} \]

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 = 0.44, size = 83, normalized size = 1.30 \[ -\frac {1}{2} \, {\left (\frac {x}{a^{4} c^{2} x^{2} + a^{2} c^{2}} - \frac {\arctan \left (a x\right )}{a^{3} c^{2}}\right )} \arctan \left (a x\right ) - \frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a}{4 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \]

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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mupad [B]  time = 0.40, size = 48, normalized size = 0.75 \[ \frac {a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2-2\,a\,x\,\mathrm {atan}\left (a\,x\right )+{\mathrm {atan}\left (a\,x\right )}^2-1}{4\,a^3\,c^2\,\left (a^2\,x^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2} \operatorname {atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

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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