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

Optimal. Leaf size=181 \[ -\frac{x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac{\tan ^{-1}(a x)}{64 a^3 c^3} \]

[Out]

x/(32*a^2*c^3*(1 + a^2*x^2)^2) - x/(64*a^2*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(64*a^3*c^3) - ArcTan[a*x]/(8*a^3*
c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(4*a^2*c^3*(1 + a^2*x^2)^2) +
(x*ArcTan[a*x]^2)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(24*a^3*c^3)

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Rubi [A]  time = 0.266443, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4964, 4892, 4930, 199, 205, 4900} \[ -\frac{x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac{x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac{\tan ^{-1}(a x)}{64 a^3 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(32*a^2*c^3*(1 + a^2*x^2)^2) - x/(64*a^2*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(64*a^3*c^3) - ArcTan[a*x]/(8*a^3*
c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(4*a^2*c^3*(1 + a^2*x^2)^2) +
(x*ArcTan[a*x]^2)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(24*a^3*c^3)

Rule 4964

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

Rule 4900

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

Rubi steps

\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac{\int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c}\\ &=-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^3}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac{\int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a c}\\ &=\frac{x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)}{2 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac{3 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c}\\ &=\frac{x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{13 x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{64 a^2 c^2}-\frac{\int \frac{1}{c+a^2 c x^2} \, dx}{4 a^2 c^2}+\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c}\\ &=\frac{x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{13 \tan ^{-1}(a x)}{64 a^3 c^3}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}+\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{16 a^2 c^2}\\ &=\frac{x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac{x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{64 a^3 c^3}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}\\ \end{align*}

Mathematica [A]  time = 0.108138, size = 95, normalized size = 0.52 \[ \frac{-3 a^3 x^3+24 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^2+8 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3-3 \left (a^4 x^4-6 a^2 x^2+1\right ) \tan ^{-1}(a x)+3 a x}{192 a^3 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x - 3*a^3*x^3 - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] + 24*a*x*(-1 + a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^
2*x^2)^2*ArcTan[a*x]^3)/(192*a^3*c^3*(1 + a^2*x^2)^2)

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Maple [A]  time = 0.041, size = 164, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{24\,{c}^{3}{a}^{3}}}-{\frac{\arctan \left ( ax \right ) }{8\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{\arctan \left ( ax \right ) }{8\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{{x}^{3}}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{x}{64\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{\arctan \left ( ax \right ) }{64\,{c}^{3}{a}^{3}}} \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)^3,x)

[Out]

1/8/c^3*arctan(a*x)^2*x^3/(a^2*x^2+1)^2-1/8*x*arctan(a*x)^2/a^2/c^3/(a^2*x^2+1)^2+1/24*arctan(a*x)^3/a^3/c^3-1
/8*arctan(a*x)/a^3/c^3/(a^2*x^2+1)^2+1/8*arctan(a*x)/a^3/c^3/(a^2*x^2+1)-1/64/c^3/(a^2*x^2+1)^2*x^3+1/64*x/a^2
/c^3/(a^2*x^2+1)^2-1/64*arctan(a*x)/a^3/c^3

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Maxima [A]  time = 1.66396, size = 313, normalized size = 1.73 \begin{align*} \frac{1}{8} \,{\left (\frac{a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac{\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac{{\left (3 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{192 \,{\left (a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}\right )}} + \frac{{\left (a^{2} x^{2} -{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )}{8 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\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)^3,x, algorithm="maxima")

[Out]

1/8*((a^2*x^3 - x)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3) + arctan(a*x)/(a^3*c^3))*arctan(a*x)^2 - 1/192*(3*a
^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x + 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a^2/(a^9
*c^3*x^4 + 2*a^7*c^3*x^2 + a^5*c^3) + 1/8*(a^2*x^2 - (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a*arctan(a*x)/(a
^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)

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Fricas [A]  time = 2.14738, size = 255, normalized size = 1.41 \begin{align*} -\frac{3 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 24 \,{\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{2} - 3 \, a x + 3 \,{\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{192 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\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)^3,x, algorithm="fricas")

[Out]

-1/192*(3*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 24*(a^3*x^3 - a*x)*arctan(a*x)^2 - 3*a*x + 3*(
a^4*x^4 - 6*a^2*x^2 + 1)*arctan(a*x))/(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)

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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^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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)**3,x)

[Out]

Integral(x**2*atan(a*x)**2/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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