\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)^3}{x^2} \, dx\) [433]

Optimal result

Integrand size = 24, antiderivative size = 1027 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^2} \, dx=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {6 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}} \]

[Out]

-1/4*a*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+1/4*a^2*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3+45/8*I*a*c^3*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^2*c*x^2+c)^(1/2)-11*I*a*c^3*arctan(a*x)*a
rctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*a*c^3*arctan(a*x)^2*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)-11/2*I*a*c^3*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*
a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+11/2*I*a*c^3*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(
a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-45/4*I*a*c^3*polylog(4,-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)-45/8*I*a*c^3*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^2
*c*x^2+c)^(1/2)+45/4*I*a*c^3*polylog(4,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*
I*a*c^3*arctan(a*x)*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)-6*a*c^3*polyl
og(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)-45/4*a*c^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^2*c*x^2+c)^(1/2)+45/4*a*c^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^2*c*x^2+c)^(1/2)+6*a*c^3*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*a*c^3*arctan(a*x)*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)-15/4*I*a*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)/(a^2*c
*x^2+c)^(1/2)-1/4*a*c^2*(a^2*c*x^2+c)^(1/2)+1/4*a^2*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-21/8*a*c^2*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)/x+7/8*a^2*c^2*x*arctan(a*x)^3*(a^2*c*x^2+c)^(1/
2)

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.00, number of steps used = 56, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5070, 5064, 5078, 5076, 4268, 2611, 2320, 6724, 5010, 5008, 4266, 6744, 5000, 5006, 4998} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^2} \, dx=-\frac {15 i a \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3 c^3}{4 \sqrt {a^2 c x^2+c}}-\frac {11 i a \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {6 i a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {45 i a \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right ) c^3}{8 \sqrt {a^2 c x^2+c}}-\frac {45 i a \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right ) c^3}{8 \sqrt {a^2 c x^2+c}}-\frac {6 i a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {11 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{2 \sqrt {a^2 c x^2+c}}-\frac {11 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{2 \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {45 a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {45 a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {6 a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {45 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {45 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {7}{8} a^2 x \sqrt {a^2 c x^2+c} \arctan (a x)^3 c^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 c^2}{x}-\frac {21}{8} a \sqrt {a^2 c x^2+c} \arctan (a x)^2 c^2+\frac {1}{4} a^2 x \sqrt {a^2 c x^2+c} \arctan (a x) c^2-\frac {1}{4} a \sqrt {a^2 c x^2+c} c^2+\frac {1}{4} a^2 x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3 c-\frac {1}{4} a \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2 c \]

[In]

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

[Out]

-1/4*(a*c^2*Sqrt[c + a^2*c*x^2]) + (a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (21*a*c^2*Sqrt[c + a^2*c*x^
2]*ArcTan[a*x]^2)/8 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x
+ (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((15*
I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((11*I)*a*c^3*Sqr
t[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^
2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a
*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Po
lyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Poly
Log[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*
ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqr
t[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1
- I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^
2] - (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (45
*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (6*a*c^3*Sqrt[
1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[
4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan
[a*x])])/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 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 4998

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*((d + e*x^2)^q/(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]), x], x] + Simp[x*(d
+ e*x^2)^q*((a + b*ArcTan[c*x])/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 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 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 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -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^2} \, dx+\left (a^2 c\right ) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx \\ & = -\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx+\frac {1}{2} \left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+\frac {1}{4} \left (3 a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx+\left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx \\ & = -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+c^3 \int \frac {\arctan (a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (a^2 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c^3\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (9 a^2 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\left (3 a c^3\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{\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}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3267\) vs. \(2(1027)=2054\).

Time = 15.01 (sec) , antiderivative size = 3267, normalized size of antiderivative = 3.18 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

((-I)*a*c^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])]))/Sqrt[1 + a^2*x^2] + (a*c^2*Sqrt[c*(1 + a^2*x^2)]*Cs
c[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((24*
I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x
^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sq
rt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1
+ I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*
x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTa
n[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a
*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*Arc
Tan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi
^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*Arc
Tan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])])/Sqrt[1 +
a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi^2*x*PolyL
og[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x
])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((
384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)
/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384
*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[
1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*PolyLog[3
, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] -
((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan[a*x]/2])/(128*Sqrt[1 + a^2*x^2
]) + a*c^2*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(Arc
Tan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])
] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/
2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/
2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4
+ (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) +
(2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])
+ 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2
- ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 -
ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a
*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (
(3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + Ar
cTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (
I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])
/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi
/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2
)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E
^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2
- (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)
/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2
))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2]
- Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1
+ a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x
]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3
)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Si
n[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(
-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2
])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(
Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[Ar
cTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])))

Maple [A] (verified)

Time = 7.82 (sec) , antiderivative size = 655, normalized size of antiderivative = 0.64

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 a^{4} \arctan \left (a x \right )^{3} x^{4}-2 a^{3} \arctan \left (a x \right )^{2} x^{3}+9 \arctan \left (a x \right )^{3} x^{2} a^{2}+2 a^{2} \arctan \left (a x \right ) x^{2}-23 a \arctan \left (a x \right )^{2} x -8 \arctan \left (a x \right )^{3}-2 a x \right )}{8 x}+\frac {i c^{2} a \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 i \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-90 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+48 i \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-48 i \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-45 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+45 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+15 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 )-15 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 )+90 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+44 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-48 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+48 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-24 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-44 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-44 \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+44 \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 \sqrt {a^{2} x^{2}+1}}\)	\(655\)

[In]

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

[Out]

1/8*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(2*a^4*arctan(a*x)^3*x^4-2*a^3*arctan(a*x)^2*x^3+9*arctan(a*x)^3*x^2*a^2+2*a
^2*arctan(a*x)*x^2-23*a*arctan(a*x)^2*x-8*arctan(a*x)^3-2*a*x)/x+1/8*I*c^2*a*(c*(a*x-I)*(I+a*x))^(1/2)*(24*I*a
rctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-90*I*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+48*I*
polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-48*I*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-45*arctan(a*x)^2*polylog(2
,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+45*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+15*I*arctan(a*x)^3*
ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-15*I*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+90*I*arctan(a*x)*po
lylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+44*I*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-48*arctan(a*x)*po
lylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+48*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-24*I*arctan(a*x)^2
*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-44*I*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-44*polylog(2,I*(1+I*a*
x)/(a^2*x^2+1)^(1/2))+90*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+44*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))
-90*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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