\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)^3}{x} \, dx\) [432]

Optimal result

Integrand size = 24, antiderivative size = 845 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

1/10*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-3/20*a*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+1/3*c*(a^2*c*x^2+c)^(3/2)*
arctan(a*x)^3+1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3-3/2*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))+149/2
0*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-2*c^3*arctan(a
*x)^3*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*c^3*polylog(4,-(1+I*a*x)/
(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+3*I*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1
)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+149/20*I*c^3*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^2*c*x^2+c)^(1/2)-149/20*I*c^3*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^2*c*x^2+c)^(1/2)-6*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2
)/(a^2*c*x^2+c)^(1/2)+149/20*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/
2)-149/20*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*c^3*arctan(a*x)
*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*c^3*polylog(4,(1+I*a*x)/(a^2
*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-3*I*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/
2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/20*a*c^2*x*(a^2*c*x^2+c)^(1/2)+29/20*c^2*arctan(a*x)*(a^2*c*x^2+c)
^(1/2)-29/40*a*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+c^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.00, number of steps used = 54, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5070, 5078, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5010, 5008, 4266, 5000, 223, 212, 201} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\frac {149 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2 c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {149 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {149 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {149 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {149 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {3}{2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right ) c^{5/2}+\sqrt {a^2 c x^2+c} \arctan (a x)^3 c^2-\frac {29}{40} a x \sqrt {a^2 c x^2+c} \arctan (a x)^2 c^2+\frac {29}{20} \sqrt {a^2 c x^2+c} \arctan (a x) c^2-\frac {1}{20} a x \sqrt {a^2 c x^2+c} c^2+\frac {1}{3} \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3 c-\frac {3}{20} a x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2 c+\frac {1}{10} \left (a^2 c x^2+c\right )^{3/2} \arctan (a x) c+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3 \]

[In]

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

[Out]

-1/20*(a*c^2*x*Sqrt[c + a^2*c*x^2]) + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*A
rcTan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x
]^2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c
^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcT
an[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(
5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^
(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*A
rcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a
*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c
+ a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149
*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^
2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3,
E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c
+ a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 201

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]])

Rule 212

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

Rule 223

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 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 4268

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

Rule 5000

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] &&
EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

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 5070

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

Rule 5076

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[1/Sqrt[d], Sub
st[Int[(a + b*x)^p*Csc[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 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*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 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*} \text {integral}& = c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx \\ & = \frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {1}{5} (3 a c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx+c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \\ & = \frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {1}{10} \left (a c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx-\frac {1}{20} \left (9 a c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx-\left (a c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+c^3 \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {1}{20} \left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{40} \left (9 a c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{20} \left (9 a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\left (3 a c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {1}{20} \left (a c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )-\frac {1}{20} \left (9 a c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )-\left (a c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{40 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{40 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \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 )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (i c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (i c^3 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \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 )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \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 )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.86 (sec) , antiderivative size = 723, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (-120 i \pi ^4+960 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)-150 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)+1392 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+960 \sqrt {1+a^2 x^2} \arctan (a x)^3+640 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^3+32 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^3+240 i \arctan (a x)^4-1440 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+960 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \cos (2 \arctan (a x))-216 \left (1+a^2 x^2\right )^{5/2} \arctan (a x) \cos (2 \arctan (a x))-160 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^3 \cos (2 \arctan (a x))-66 \left (1+a^2 x^2\right )^{5/2} \arctan (a x) \cos (4 \arctan (a x))+960 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-2880 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+2880 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-960 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+2880 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+2880 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-7152 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+7152 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+5760 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-5760 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+7152 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-7152 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-5760 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-5760 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-12 \left (1+a^2 x^2\right )^{5/2} \sin (2 \arctan (a x))-480 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \sin (2 \arctan (a x))-6 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^2 \sin (2 \arctan (a x))-6 \left (1+a^2 x^2\right )^{5/2} \sin (4 \arctan (a x))+33 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^2 \sin (4 \arctan (a x))\right )}{960 \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*((-120*I)*Pi^4 + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] - 150*(1 + a^2*x^2)^(5/2)*ArcTan
[a*x] + (1392*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 960*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 640*(1 + a^2*
x^2)^(3/2)*ArcTan[a*x]^3 + 32*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3 + (240*I)*ArcTan[a*x]^4 - 1440*ArcTanh[(a*x)/S
qrt[1 + a^2*x^2]] + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 216*(1 + a^2*x^2)^(5/2)*ArcTan[a*
x]*Cos[2*ArcTan[a*x]] - 160*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 66*(1 + a^2*x^2)^(5/2)*ArcT
an[a*x]*Cos[4*ArcTan[a*x]] + 960*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 2880*ArcTan[a*x]^2*Log[1 - I*E^
(I*ArcTan[a*x])] + 2880*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 960*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x
])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan
[a*x])] - (7152*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (7152*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*Arc
Tan[a*x])] + 5760*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 5760*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x]
)] + 7152*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 7152*PolyLog[3, I*E^(I*ArcTan[a*x])] - (5760*I)*PolyLog[4, E^((
-I)*ArcTan[a*x])] - (5760*I)*PolyLog[4, -E^(I*ArcTan[a*x])] - 12*(1 + a^2*x^2)^(5/2)*Sin[2*ArcTan[a*x]] - 480*
(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] - 6*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]]
- 6*(1 + a^2*x^2)^(5/2)*Sin[4*ArcTan[a*x]] + 33*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(960*Sq
rt[1 + a^2*x^2])

Maple [A] (verified)

Time = 7.49 (sec) , antiderivative size = 562, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 a^{4} \arctan \left (a x \right )^{3} x^{4}-18 a^{3} \arctan \left (a x \right )^{2} x^{3}+88 \arctan \left (a x \right )^{3} x^{2} a^{2}+12 a^{2} \arctan \left (a x \right ) x^{2}-105 a \arctan \left (a x \right )^{2} x +184 \arctan \left (a x \right )^{3}-6 a x +186 \arctan \left (a x \right )\right )}{120}-\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (40 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-40 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-149 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+149 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+298 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-298 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+240 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+240 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-298 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+298 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40 \sqrt {a^{2} x^{2}+1}}\)	\(562\)

[In]

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

[Out]

1/120*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(24*a^4*arctan(a*x)^3*x^4-18*a^3*arctan(a*x)^2*x^3+88*arctan(a*x)^3*x^2*a^
2+12*a^2*arctan(a*x)*x^2-105*a*arctan(a*x)^2*x+184*arctan(a*x)^3-6*a*x+186*arctan(a*x))-1/40*c^2*(c*(a*x-I)*(I
+a*x))^(1/2)*(40*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-40*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(
1/2))-120*I*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2
*x^2+1)^(1/2))-149*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+149*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x
^2+1)^(1/2))+298*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-298*I*arctan(a*x)*polylog(2,I*(1+I*a*
x)/(a^2*x^2+1)^(1/2))+240*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*arctan(a*x)*polylog(3,(1+I*a
*x)/(a^2*x^2+1)^(1/2))+240*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*I*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/
2))-120*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))-298*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+298*polylog(3,I*(1
+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3/x, x)

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,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. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \]

[In]

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

[Out]

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