3.397 \(\int \frac{x^2 \tan ^{-1}(a x)^3}{(c+a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=135 \[ \frac{3}{8 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2} \]

[Out]

3/(8*a^3*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(8*a^3*c^2) - (3
*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a^3
*c^2)

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Rubi [A]  time = 0.143749, antiderivative size = 135, 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, 4892, 261} \[ \frac{3}{8 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]

[Out]

3/(8*a^3*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(8*a^3*c^2) - (3
*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*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 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{x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a}\\ &=-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2}\\ &=\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}-\frac{3 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=\frac{3}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0596125, size = 74, normalized size = 0.55 \[ \frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^4+3 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^2-4 a x \tan ^{-1}(a x)^3+6 a x \tan ^{-1}(a x)+3}{8 a^3 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.158, size = 124, normalized size = 0.9 \begin{align*}{\frac{3}{8\,{a}^{3}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\,x\arctan \left ( ax \right ) }{4\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8\,{a}^{3}{c}^{2}}}-{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{a}^{3}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{3}}{2\,{a}^{2}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{4}}{8\,{a}^{3}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^2,x)

[Out]

3/8/a^3/c^2/(a^2*x^2+1)+3/4*x*arctan(a*x)/a^2/c^2/(a^2*x^2+1)+3/8*arctan(a*x)^2/a^3/c^2-3/4*arctan(a*x)^2/a^3/
c^2/(a^2*x^2+1)-1/2*x*arctan(a*x)^3/a^2/c^2/(a^2*x^2+1)+1/8*arctan(a*x)^4/a^3/c^2

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Maxima [A]  time = 1.73454, size = 294, normalized size = 2.18 \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 )^{3} - \frac{3 \,{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a \arctan \left (a x\right )^{2}}{4 \,{\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} - \frac{1}{8} \,{\left (\frac{{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \,{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 3\right )} a^{2}}{a^{8} c^{2} x^{2} + a^{6} c^{2}} - \frac{2 \,{\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 \arctan \left (a x\right )}{a^{7} c^{2} x^{2} + a^{5} c^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(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)^3 - 3/4*((a^2*x^2 + 1)*arctan(a*x)^2 + 1)
*a*arctan(a*x)^2/(a^6*c^2*x^2 + a^4*c^2) - 1/8*(((a^2*x^2 + 1)*arctan(a*x)^4 + 3*(a^2*x^2 + 1)*arctan(a*x)^2 -
3)*a^2/(a^8*c^2*x^2 + a^6*c^2) - 2*(2*(a^2*x^2 + 1)*arctan(a*x)^3 + 3*a*x + 3*(a^2*x^2 + 1)*arctan(a*x))*a*ar
ctan(a*x)/(a^7*c^2*x^2 + a^5*c^2))*a

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Fricas [A]  time = 1.92814, size = 186, normalized size = 1.38 \begin{align*} -\frac{4 \, a x \arctan \left (a x\right )^{3} -{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 6 \, a x \arctan \left (a x\right ) - 3 \,{\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} - 3}{8 \,{\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)^3/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/8*(4*a*x*arctan(a*x)^3 - (a^2*x^2 + 1)*arctan(a*x)^4 - 6*a*x*arctan(a*x) - 3*(a^2*x^2 - 1)*arctan(a*x)^2 -
3)/(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}^{3}{\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)**3/(a**2*c*x**2+c)**2,x)

[Out]

Integral(x**2*atan(a*x)**3/(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 )^{3}}{{\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)^3/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2 + c)^2, x)