\(\int \frac {(c+a^2 c x^2) \arctan (a x)^3}{x^2} \, dx\) [368]

Optimal result

Integrand size = 20, antiderivative size = 169 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]

[Out]

-c*arctan(a*x)^3/x+a^2*c*x*arctan(a*x)^3+3*a*c*arctan(a*x)^2*ln(2/(1+I*a*x))+3*a*c*arctan(a*x)^2*ln(2-2/(1-I*a
*x))-3*I*a*c*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+3*I*a*c*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+3/2*a*c*polylo
g(3,-1+2/(1-I*a*x))+3/2*a*c*polylog(3,1-2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5070, 4946, 5044, 4988, 5004, 5112, 6745, 4930, 5040, 4964, 5114} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=a^2 c x \arctan (a x)^3-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-\frac {c \arctan (a x)^3}{x}+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right ) \]

[In]

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

[Out]

-((c*ArcTan[a*x]^3)/x) + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c*ArcTan[a*x]^2*
Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a*c*ArcTan[a*x]*PolyLog[
2, 1 - 2/(1 + I*a*x)] + (3*a*c*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2

Rule 4930

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

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4964

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

Rule 4988

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

Rule 5004

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

Rule 5040

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

Rule 5044

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

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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] - Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]
/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I
/(I + c*x)))^2, 0]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2
*(I/(I - c*x)))^2, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\arctan (a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \arctan (a x)^3 \, dx \\ & = -\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+(3 a c) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^3 c\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+(3 i a c) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx+\left (3 a^2 c\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx \\ & = -\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-\left (6 a^2 c\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^2 c\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\left (3 i a^2 c\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^2 c\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-i a c \arctan (a x)^3+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+a c \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right ) \]

[In]

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

[Out]

(-I)*a*c*ArcTan[a*x]^3 + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*a*
c*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + a*c*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x)
+ 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*
PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2) + (3*a*c*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 14.21 (sec) , antiderivative size = 1653, normalized size of antiderivative = 9.78

method	result	size
parts	\(\text {Expression too large to display}\)	\(1653\)
derivativedivides	\(\text {Expression too large to display}\)	\(1654\)
default	\(\text {Expression too large to display}\)	\(1654\)

[In]

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

[Out]

a^2*c*x*arctan(a*x)^3-c*arctan(a*x)^3/x-3*c*(a*arctan(a*x)^2*ln(a^2*x^2+1)-a*arctan(a*x)^2*ln(a*x)-2*a*(arctan
(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/3*I*arctan(a*x)^3+1/4*(I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I
*a*x)^2/(a^2*x^2+1)+1))^3-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*cs
gn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1
)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+I*Pi*csg
n(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*
a*x)^2/(a^2*x^2+1)+1))-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+I*Pi*csgn(((
1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+
I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-I*Pi*csgn(I*((1+I*a*x)^2/(
a^2*x^2+1)-1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2
*x^2+1)+1)^2)^3+I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1
)^2)^2-I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+
1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+I*Pi+I*Pi*c
sgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-2*I*Pi*cs
gn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)
-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))+2*I*Pi*csgn(I*(
1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2+4*ln(2))*arctan(a*x)^2-1/2*arctan(a*x)^2*ln((1+I
*a*x)^2/(a^2*x^2+1)-1)+1/2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-I*arctan(a*x)*polylog(2,-(1+I*a*x)/
(a^2*x^2+1)^(1/2))+polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/2*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))
-I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/2*I*arctan(a*x)
*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+1/4*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/64*(8*(a^2*c*x^2 - c)*arctan(a*x)^3 - 6*(a^2*c*x^2 - c)*arctan(a*x)*log(a^2*x^2 + 1)^2 + (28*a*c*arctan(a*x)
^4 + 1792*a^4*c*integrate(1/32*x^4*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 192*a^4*c*integrate(1/32*x^4*arctan(a*x
)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 768*a^4*c*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4
+ x^2), x) - 768*a^3*c*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + a*c*log(a^2*x^2 + 1)^3 + 384*a^2
*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 768*a^2*c*integrate(1/32*x^2*arctan
(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 768*a*c*integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 192*
a*c*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 1792*c*integrate(1/32*arctan(a*x)^3/(a^2*x^4 + x
^2), x) + 192*c*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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