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

Optimal. Leaf size=67 \[ \frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}-\frac{1}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{1}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.279959, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4964, 4902, 4970, 4406, 12, 3299} \[ \frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}-\frac{1}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{1}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 1/(a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[4*ArcTan[a*x]]/(
2*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 4902

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

Rule 4970

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(m
+ 1), Subst[Int[((a + b*x)^p*Sin[x]^m)/Cos[x]^(m + 2*(q + 1)), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d,
e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=\frac{1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{4 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}-\frac{2 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a c}\\ &=\frac{1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac{1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac{4 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 x}+\frac{\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac{1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^3}\\ &=\frac{1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}\\ \end{align*}

Mathematica [A]  time = 0.170959, size = 59, normalized size = 0.88 \[ \frac{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text{Si}\left (4 \tan ^{-1}(a x)\right )-2 a^2 x^2}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.071, size = 37, normalized size = 0.6 \begin{align*}{\frac{4\,{\it Si} \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +\cos \left ( 4\,\arctan \left ( ax \right ) \right ) -1}{8\,{a}^{3}{c}^{3}\arctan \left ( ax \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right ) \int \frac{a^{2} x^{3} - x}{{\left (a^{7} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )}\,{d x} + x^{2}}{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x)*integrate(2*(a^2*x^3 - x)/((a^7*c^3*x^6 + 3*a^5*c^3*x^4 +
3*a^3*c^3*x^2 + a*c^3)*arctan(a*x)), x) + x^2)/((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x))

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Fricas [C]  time = 2.08475, size = 460, normalized size = 6.87 \begin{align*} -\frac{4 \, a^{2} x^{2} -{\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \logintegral \left (\frac{a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) -{\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \logintegral \left (\frac{a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )} \arctan \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="fricas")

[Out]

-1/4*(4*a^2*x^2 - (I*a^4*x^4 + 2*I*a^2*x^2 + I)*arctan(a*x)*log_integral((a^4*x^4 + 4*I*a^3*x^3 - 6*a^2*x^2 -
4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)) - (-I*a^4*x^4 - 2*I*a^2*x^2 - I)*arctan(a*x)*log_integral((a^4*x^4 - 4
*I*a^3*x^3 - 6*a^2*x^2 + 4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)))/((a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)*arc
tan(a*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{2}}{a^{6} x^{6} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^2*c*x^2+c)^3/arctan(a*x)^2,x, algorithm="giac")

[Out]

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