3.430 \(\int x (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=561 \[ -\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{17 c^2 x \sqrt{a^2 c x^2+c}}{420 a}-\frac{15 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac{15 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{120 a^2}-\frac{c x \left (a^2 c x^2+c\right )^{3/2}}{140 a}+\frac{\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{5 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac{5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{84 a^2} \]

[Out]

(-17*c^2*x*Sqrt[c + a^2*c*x^2])/(420*a) - (c*x*(c + a^2*c*x^2)^(3/2))/(140*a) + (15*c^2*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a^2) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(
35*a^2) - (15*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(
56*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(14*a) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan
[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^3)/(7*a^2*c) - (37*c^(5/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^2) - (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Pol
yLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTa
n[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(56*a^2*Sq
rt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.531382, antiderivative size = 561, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ -\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{17 c^2 x \sqrt{a^2 c x^2+c}}{420 a}-\frac{15 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac{15 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{120 a^2}-\frac{c x \left (a^2 c x^2+c\right )^{3/2}}{140 a}+\frac{\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{5 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac{5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{84 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(-17*c^2*x*Sqrt[c + a^2*c*x^2])/(420*a) - (c*x*(c + a^2*c*x^2)^(3/2))/(140*a) + (15*c^2*Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a^2) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(
35*a^2) - (15*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(
56*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(14*a) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan
[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^3)/(7*a^2*c) - (37*c^(5/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^2) - (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Pol
yLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTa
n[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(56*a^2*Sq
rt[c + a^2*c*x^2])

Rule 4930

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 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 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{3 \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx}{7 a}\\ &=\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{c \int \left (c+a^2 c x^2\right )^{3/2} \, dx}{35 a}-\frac{(5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx}{14 a}\\ &=-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{\left (3 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx}{140 a}-\frac{\left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx}{84 a}-\frac{\left (15 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx}{56 a}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{\left (3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{280 a}-\frac{\left (5 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{168 a}-\frac{\left (15 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{112 a}-\frac{\left (15 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{56 a}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{280 a}-\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{168 a}-\frac{\left (15 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{56 a}-\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{112 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^2}-\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{112 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^2}+\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^2}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (15 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (15 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^2}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 x \sqrt{c+a^2 c x^2}}{420 a}-\frac{c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{15 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac{5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{37 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^2}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}+\frac{15 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}-\frac{15 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 5.54493, size = 718, normalized size = 1.28 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (64 \left (-309 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+309 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+309 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-309 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-259 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+309 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+53760 \left (-i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-\tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )-112 \left (\left (a^2 x^2+1\right )^{5/2} \left (\frac{48 a x}{\left (a^2 x^2+1\right )^2}+\tan ^{-1}(a x)^2 \left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right )+32 \tan ^{-1}(a x)^3 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right )+6 \tan ^{-1}(a x) \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )\right )+48 \left (-11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+11 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-11 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-10 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )\right )+\left (a^2 x^2+1\right )^{7/2} \left (\frac{8 \tan ^{-1}(a x) \left (764 \cos \left (2 \tan ^{-1}(a x)\right )+309 \cos \left (4 \tan ^{-1}(a x)\right )+647\right )}{a^2 x^2+1}-3 \tan ^{-1}(a x)^2 \left (211 \sin \left (2 \tan ^{-1}(a x)\right )-60 \sin \left (4 \tan ^{-1}(a x)\right )+103 \sin \left (6 \tan ^{-1}(a x)\right )\right )+4 \left (101 \sin \left (2 \tan ^{-1}(a x)\right )+88 \sin \left (4 \tan ^{-1}(a x)\right )+25 \sin \left (6 \tan ^{-1}(a x)\right )\right )+64 \tan ^{-1}(a x)^3 \left (-28 \cos \left (2 \tan ^{-1}(a x)\right )+35 \cos \left (4 \tan ^{-1}(a x)\right )+57\right )\right )+4480 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \left (4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+6 \cos \left (2 \tan ^{-1}(a x)\right )+6\right )\right )}{53760 a^2 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2
*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan
[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 53760*(I*ArcTan[E^(I
*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*
x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*
ArcTan[a*x])]) + 4480*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a
*x]*Sin[2*ArcTan[a*x]]) - 112*(48*((11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 10*ArcTanh[(a*x)/Sqrt[1 +
a^2*x^2]] - (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTa
n[a*x])] + 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(5/2)*(
(48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[
a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]]))) + (1 + a^2*x^2
)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos
[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] +
25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a
*x]]))))/(53760*a^2*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 1.486, size = 477, normalized size = 0.9 \begin{align*}{\frac{{c}^{2} \left ( 240\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{6}{a}^{6}-120\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}+720\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}{a}^{4}+48\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-390\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+720\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}-12\,{a}^{3}{x}^{3}+196\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-495\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+240\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-80\,ax+598\,\arctan \left ( ax \right ) \right ) }{1680\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{5\,{c}^{2}}{112\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}-3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{5\,{c}^{2}}{112\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}+6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\frac{37\,i}{60}}{c}^{2}}{{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x)

[Out]

1/1680*c^2/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(240*arctan(a*x)^3*x^6*a^6-120*arctan(a*x)^2*x^5*a^5+720*arctan(a*x)^
3*x^4*a^4+48*arctan(a*x)*x^4*a^4-390*arctan(a*x)^2*x^3*a^3+720*arctan(a*x)^3*x^2*a^2-12*a^3*x^3+196*arctan(a*x
)*a^2*x^2-495*arctan(a*x)^2*x*a+240*arctan(a*x)^3-80*a*x+598*arctan(a*x))-5/112*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*
(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a
^2*x^2+1)^(1/2))-6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+5/112*c^2*(c*(a*x-I)*(a*x+
I))^(1/2)*(I*arctan(a*x)^3+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1-I*(1+
I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^2/(a^2*x^2+1)^(1/2)+37/60*I*c^2/a^2*(c
*(a*x-I)*(a*x+I))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*x*arctan(a*x)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**(5/2)*atan(a*x)**3,x)

[Out]

Timed out

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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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError