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

Optimal. Leaf size=626 \[ \frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}} \]

[Out]

(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (I*c*Sqrt[1 + a^2*x^2
]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Ar
cTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Pol
yLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I
*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/S
qrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x]
)])/(a*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c +
a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*
c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.361453, antiderivative size = 626, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886} \[ \frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (I*c*Sqrt[1 + a^2*x^2
]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Ar
cTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Pol
yLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I
*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/S
qrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x]
)])/(a*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c +
a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*
c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])

Rule 4880

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

Rule 4890

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 4888

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 4181

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 2531

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 6609

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]

Rule 2282

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 6589

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 4886

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]

Rubi steps

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

Mathematica [A]  time = 0.798037, size = 258, normalized size = 0.41 \[ -\frac{i \sqrt{c \left (a^2 x^2+1\right )} \left (-6 i \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+6 i \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-3 \left (\tan ^{-1}(a x)^2+2\right ) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+3 \left (\tan ^{-1}(a x)^2+2\right ) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+6 \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )-6 \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )+i a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3+12 \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)\right )}{2 a \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/2)*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^
2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]
*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^
(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 2.117, size = 422, normalized size = 0.7 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2} \left ( \arctan \left ( ax \right ) xa-3 \right ) }{2\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{1}{2\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) - \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,\arctan \left ( ax \right ) \ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i{\it polylog} \left ( 4,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i{\it polylog} \left ( 4,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError