3.5.37 \(\int \frac {x^2 \text {ArcTan}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\) [437]

Optimal. Leaf size=625 \[ -\frac {3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5072, 5050, 5010, 5006, 5008, 4266, 2611, 6744, 2320, 6724} \begin {gather*} \frac {x \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {3 i \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \text {ArcTan}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{2 a^3 c}-\frac {6 i \sqrt {a^2 x^2+1} \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \text {ArcTan}(a x)}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) + (I*Sqrt[1
+ a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcTan[
a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*A
rcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a
^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2
]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])

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 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 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 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 5072

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + (-Dist[b*f*(p/(c*m)), Int[(f*x)^(m - 1
)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[f^2*((m - 1)/(c^2*m)), Int[(f*x)^(m - 2)*((a +
b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && Gt
Q[m, 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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac {\int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {3 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.26, size = 812, normalized size = 1.30 \begin {gather*} \frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\frac {7 i \pi ^4}{32}+\frac {1}{4} i \pi ^3 \text {ArcTan}(a x)-6 \text {ArcTan}(a x)^2-\frac {3}{4} i \pi ^2 \text {ArcTan}(a x)^2+i \pi \text {ArcTan}(a x)^3-\frac {1}{2} i \text {ArcTan}(a x)^4-\frac {3}{2} \pi ^2 \text {ArcTan}(a x) \log \left (1-i e^{-i \text {ArcTan}(a x)}\right )+3 \pi \text {ArcTan}(a x)^2 \log \left (1-i e^{-i \text {ArcTan}(a x)}\right )+\frac {1}{4} \pi ^3 \log \left (1+i e^{-i \text {ArcTan}(a x)}\right )-2 \text {ArcTan}(a x)^3 \log \left (1+i e^{-i \text {ArcTan}(a x)}\right )+12 \text {ArcTan}(a x) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-\frac {1}{4} \pi ^3 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-12 \text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+\frac {3}{2} \pi ^2 \text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-3 \pi \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+2 \text {ArcTan}(a x)^3 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-\frac {1}{4} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-6 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcTan}(a x)}\right )-\frac {3}{2} i \pi (\pi -4 \text {ArcTan}(a x)) \text {PolyLog}\left (2,i e^{-i \text {ArcTan}(a x)}\right )+12 i \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-\frac {3}{2} i \pi ^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+6 i \pi \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-6 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-12 i \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-12 \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{-i \text {ArcTan}(a x)}\right )+6 \pi \text {PolyLog}\left (3,i e^{-i \text {ArcTan}(a x)}\right )-6 \pi \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+12 \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+12 i \text {PolyLog}\left (4,-i e^{-i \text {ArcTan}(a x)}\right )+12 i \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )+\frac {\text {ArcTan}(a x)^3}{\left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )^2}-\frac {6 \text {ArcTan}(a x)^2 \sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )}{\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )}-\frac {\text {ArcTan}(a x)^3}{\left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )^2}+\frac {6 \text {ArcTan}(a x)^2 \sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )}{\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )}\right )}{4 a^3 c \sqrt {1+a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*(1 + a^2*x^2)]*(((7*I)/32)*Pi^4 + (I/4)*Pi^3*ArcTan[a*x] - 6*ArcTan[a*x]^2 - ((3*I)/4)*Pi^2*ArcTan[a*x
]^2 + I*Pi*ArcTan[a*x]^3 - (I/2)*ArcTan[a*x]^4 - (3*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/2 + 3*Pi*Ar
cTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] + (Pi^3*Log[1 + I/E^(I*ArcTan[a*x])])/4 - 2*ArcTan[a*x]^3*Log[1 + I/E
^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - (Pi^3*Log[1 + I*E^(I*ArcTan[a*x])])/4 - 12*A
rcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + (3*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/2 - 3*Pi*ArcTan[a*
x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - (Pi^3*Log[Tan[(Pi + 2*ArcTa
n[a*x])/4]])/4 - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi*(Pi - 4*ArcTan[a*x])*Po
lyLog[2, I/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi^2*PolyLog[2, (-I)*E^(
I*ArcTan[a*x])] + (6*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I
)*E^(I*ArcTan[a*x])] - (12*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x
])] + 6*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 6*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*PolyLog[
3, (-I)*E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*
x])] + ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2 - (6*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(Cos[A
rcTan[a*x]/2] - Sin[ArcTan[a*x]/2]) - ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2 + (6*ArcTan[a*
x]^2*Sin[ArcTan[a*x]/2])/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])))/(4*a^3*c*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

Maple [A]
time = 7.14, size = 430, normalized size = 0.69

method	result	size
default	\(\frac {\left (\arctan \left (a x \right ) a x -3\right ) \arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}-\frac {i \left (i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c}\)	\(430\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________