\(\int \frac {x^3 \arctan (a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [444]

Optimal result

Integrand size = 24, antiderivative size = 403 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}} \]

[Out]

6*x/a^3/c/(a^2*c*x^2+c)^(1/2)-6*arctan(a*x)/a^4/c/(a^2*c*x^2+c)^(1/2)-3*x*arctan(a*x)^2/a^3/c/(a^2*c*x^2+c)^(1
/2)+arctan(a*x)^3/a^4/c/(a^2*c*x^2+c)^(1/2)+6*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^
(1/2)/a^4/c/(a^2*c*x^2+c)^(1/2)-6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^
4/c/(a^2*c*x^2+c)^(1/2)+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/c/(a^2*
c*x^2+c)^(1/2)+6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/c/(a^2*c*x^2+c)^(1/2)-6*polyl
og(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/c/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^3*(a^2*c*x^2+c)^(1
/2)/a^4/c^2

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5084, 5050, 5010, 5008, 4266, 2611, 2320, 6724, 5018, 197} \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^4 c^2}-\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 x}{a^3 c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

(6*x)/(a^3*c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (3*x*ArcTan[a*x]^2)/(a^3*c*S
qrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*c*Sqrt[c + a^2*c*
x^2]) + ArcTan[a*x]^3/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^4*c^2) - ((6*I)*Sqr
t[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a
^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLo
g[3, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x]
)])/(a^4*c*Sqrt[c + a^2*c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5008

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] &
& GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 5018

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

Rule 5050

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 5084

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c} \\ & = \frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c} \\ & = -\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}+\frac {6 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^3 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (\frac {6 a x}{\sqrt {1+a^2 x^2}}-3 \sqrt {1+a^2 x^2} \arctan (a x)-\frac {3 a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+\frac {3}{2} \sqrt {1+a^2 x^2} \arctan (a x)^3-3 \sqrt {1+a^2 x^2} \arctan (a x) \cos (2 \arctan (a x))+\frac {1}{2} \sqrt {1+a^2 x^2} \arctan (a x)^3 \cos (2 \arctan (a x))-3 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+3 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-6 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+6 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+6 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-6 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{a^4 c \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

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

[Out]

(Sqrt[1 + a^2*x^2]*((6*a*x)/Sqrt[1 + a^2*x^2] - 3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (3*a*x*ArcTan[a*x]^2)/Sqrt[1
+ a^2*x^2] + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3)/2 - 3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + (Sq
rt[1 + a^2*x^2]*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]])/2 - 3*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 3*ArcTan[
a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]
*PolyLog[2, I*E^(I*ArcTan[a*x])] + 6*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[3, I*E^(I*ArcTan[a*x])]))/
(a^4*c*Sqrt[c*(1 + a^2*x^2)])

Maple [F]

\[\int \frac {x^{3} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^3*arctan(a*x)^3/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)

Sympy [F]

\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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