\(\int \frac {(c+a^2 c x^2) \arctan (a x)^2}{x^3} \, dx\) [264]

Optimal result

Integrand size = 20, antiderivative size = 196 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=-\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c \log (x)-\frac {1}{2} a^2 c \log \left (1+a^2 x^2\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5070, 4946, 5038, 272, 36, 29, 31, 5004, 4942, 5108, 5114, 6745} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )-\frac {1}{2} a^2 c \log \left (a^2 x^2+1\right )+a^2 c \log (x)-\frac {c \arctan (a x)^2}{2 x^2}-\frac {a c \arctan (a x)}{x} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4942

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 +
I*c*x)], x] - Dist[2*b*c*p, Int[(a + b*ArcTan[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x
] /; FreeQ[{a, b, c}, x] && IGtQ[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 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 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[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 5108

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[1 + u]*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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)^2}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\arctan (a x)^2}{x} \, dx \\ & = -\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+(a c) \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (4 a^3 c\right ) \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+(a c) \int \frac {\arctan (a x)}{x^2} \, dx-\left (a^3 c\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx+\left (2 a^3 c\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a^3 c\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (a^2 c\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (i a^3 c\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a^3 c\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c \log (x)-\frac {1}{2} a^2 c \log \left (1+a^2 x^2\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=-\frac {a c \arctan (a x)}{x}+\frac {c \left (-1-a^2 x^2\right ) \arctan (a x)^2}{2 x^2}+\frac {2}{3} i a^2 c \arctan (a x)^3+a^2 c \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-a^2 c \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+a^2 c \log (x)-\frac {1}{2} a^2 c \log \left (1+a^2 x^2\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right ) \]

[In]

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

[Out]

-((a*c*ArcTan[a*x])/x) + (c*(-1 - a^2*x^2)*ArcTan[a*x]^2)/(2*x^2) + ((2*I)/3)*a^2*c*ArcTan[a*x]^3 + a^2*c*ArcT
an[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - a^2*c*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + a^2*c*Log[x]
- (a^2*c*Log[1 + a^2*x^2])/2 + I*a^2*c*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + I*a^2*c*ArcTan[a*x]*Po
lyLog[2, -E^((2*I)*ArcTan[a*x])] + (a^2*c*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 - (a^2*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 = 45.98 (sec) , antiderivative size = 1134, normalized size of antiderivative = 5.79

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1134\)
default	\(\text {Expression too large to display}\)	\(1134\)
parts	\(\text {Expression too large to display}\)	\(1561\)

[In]

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

[Out]

a^2*(c*arctan(a*x)^2*ln(a*x)-1/2*c*arctan(a*x)^2/a^2/x^2-c*(arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-arctan
(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+1/2*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*arctan(a*x)^2-2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-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))*arctan(a*x)^2-2*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*arctan(a*x)*polylog(2,-(1+I*a
*x)^2/(a^2*x^2+1))+1/2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+1/2*arctan(a*x)^2+2*I*arctan(a*x)*polylog(2,(1+I*a*
x)/(a^2*x^2+1)^(1/2))-1/2*I*Pi*arctan(a*x)^2-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-1/2*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))^3*arctan(a*x)^2-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+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))^3*arctan(a*x)^2-1/2*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))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan
(a*x)^2+1/2*arctan(a*x)*(I*a*x-(a^2*x^2+1)^(1/2)+1)/a/x+1/2*arctan(a*x)*(I*a*x+(a^2*x^2+1)^(1/2)+1)/a/x+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*arcta
n(a*x)^2+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*arctan(a*x)^2+2*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+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*arc
tan(a*x)^2))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/96*((1152*a^4*c*integrate(1/16*x^4*arctan(a*x)^2/(a^2*x^5 + x^3), x) + a^2*c*log(a^2*x^2 + 1)^3 + 2304*a^2*c
*integrate(1/16*x^2*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 192*a^2*c*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*x
^5 + x^3), x) - 192*a^2*c*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) + 384*a*c*integrate(1/16*x*a
rctan(a*x)/(a^2*x^5 + x^3), x) + 1152*c*integrate(1/16*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 96*c*integrate(1/16
*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x))*x^2 - 12*c*arctan(a*x)^2 + 3*c*log(a^2*x^2 + 1)^2)/x^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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