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

Optimal. Leaf size=177 \[ -\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{x}{2 a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{3}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{2}{a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.642054, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4964, 4932, 4970, 4406, 12, 3299, 4968, 4902} \[ -\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{x}{2 a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{3}{2 a^4 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{2}{a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - x/(2*a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + 2/(a^4*c^3*(1 + a^2*
x^2)^2*ArcTan[a*x]) - 3/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (1 - a^2*x^2)/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[
a*x]) - SinIntegral[2*ArcTan[a*x]]/(2*a^4*c^3) + SinIntegral[4*ArcTan[a*x]]/(a^4*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 4932

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

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]

Rule 4968

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

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]

Rubi steps

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

Mathematica [A]  time = 0.233105, size = 72, normalized size = 0.41 \[ \frac{\frac{a^2 x^2 \left (\left (a^2 x^2-3\right ) \tan ^{-1}(a x)-a x\right )}{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}-\text{Si}\left (2 \tan ^{-1}(a x)\right )+2 \text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.069, size = 90, normalized size = 0.5 \begin{align*} -{\frac{8\,{\it Si} \left ( 2\,\arctan \left ( ax \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}-16\,{\it Si} \left ( 4\,\arctan \left ( ax \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}+4\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -4\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +2\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) -\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{16\,{c}^{3}{a}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/a^4/c^3*(8*Si(2*arctan(a*x))*arctan(a*x)^2-16*Si(4*arctan(a*x))*arctan(a*x)^2+4*cos(2*arctan(a*x))*arcta
n(a*x)-4*cos(4*arctan(a*x))*arctan(a*x)+2*sin(2*arctan(a*x))-sin(4*arctan(a*x)))/arctan(a*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.74747, size = 799, normalized size = 4.51 \begin{align*} -\frac{2 \, a^{3} x^{3} -{\left (2 i \, a^{4} x^{4} + 4 i \, a^{2} x^{2} + 2 i\right )} \arctan \left (a x\right )^{2} \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 (-2 i \, a^{4} x^{4} - 4 i \, a^{2} x^{2} - 2 i\right )} \arctan \left (a x\right )^{2} \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 )^{2} \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{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 )^{2} \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \,{\left (a^{4} x^{4} - 3 \, a^{2} x^{2}\right )} \arctan \left (a x\right )}{4 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a^3*x^3 - (2*I*a^4*x^4 + 4*I*a^2*x^2 + 2*I)*arctan(a*x)^2*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)) - (-2*I*a^4*x^4 - 4*I*a^2*x^2 - 2*I)*arctan(a*x)^2*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)^2*log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2*x^2 + 1)) - (I*a^4*x^4 + 2*I*a^2*x^2 + I)*arctan(a*x)
^2*log_integral(-(a^2*x^2 - 2*I*a*x - 1)/(a^2*x^2 + 1)) - 2*(a^4*x^4 - 3*a^2*x^2)*arctan(a*x))/((a^8*c^3*x^4 +
2*a^6*c^3*x^2 + a^4*c^3)*arctan(a*x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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