Integrand size = 20, antiderivative size = 196 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=-\frac {a c \arctan (a x)}{x}-\frac {1}{2} a^2 c \arctan (a x)^2-\frac {c \arctan (a x)^2}{2 x^2}+2 a^2 c \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c \log (x)-\frac {1}{2} a^2 c \log \left (1+a^2 x^2\right )-i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]
-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)*polylog( 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))
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=-\frac {a c \arctan (a x)}{x}+\frac {c \left (-1-a^2 x^2\right ) \arctan (a x)^2}{2 x^2}+\frac {2}{3} i a^2 c \arctan (a x)^3+a^2 c \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-a^2 c \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+a^2 c \log (x)-\frac {1}{2} a^2 c \log \left (1+a^2 x^2\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+i a^2 c \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\frac {1}{2} a^2 c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right ) \]
-((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*ArcTan[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)*ArcT an[a*x])] + I*a^2*c*ArcTan[a*x]*PolyLog[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)*ArcT an[a*x])])/2
Time = 1.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5485, 5357, 5361, 5453, 5361, 243, 47, 14, 16, 5419, 5523, 5529, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {\arctan (a x)^2}{x}dx+c \int \frac {\arctan (a x)^2}{x^3}dx\) |
| \(\Big \downarrow \) 5357 |
| \(\displaystyle c \int \frac {\arctan (a x)^2}{x^3}dx+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (a \int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle c \left (a \left (\int \frac {\arctan (a x)}{x^2}dx-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle c \left (a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle c \left (a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle c \left (a \left (a^2 \left (-\int \frac {\arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle c \left (a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5523 |
| \(\displaystyle c \left (a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \left (\frac {1}{2} \int \frac {\arctan (a x) \log \left (2-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {1}{2} \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle c \left (a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \left (\frac {1}{2} \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle c \left (a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}\right )-\frac {\arctan (a x)^2}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 a \left (\frac {1}{2} \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )+\frac {1}{2} \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{4 a}\right )\right )\right )\) |
c*(-1/2*ArcTan[a*x]^2/x^2 + a*(-(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a *(Log[x^2] - Log[1 + a^2*x^2]))/2)) + a^2*c*(2*ArcTan[a*x]^2*ArcTanh[1 - 2 /(1 + I*a*x)] - 4*a*((((I/2)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a))/2 + (((-1/2*I)*ArcTan[a*x]*PolyLog[ 2, -1 + 2/(1 + I*a*x)])/a - PolyLog[3, -1 + 2/(1 + I*a*x)]/(4*a))/2))
3.3.64.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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] - Simp[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]
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] - Simp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> 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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[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]))
Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTan[c*x])^p/(d + e *x^2)), x], x] - Simp[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]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[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]
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]
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 | \(\text {Expression too large to display}\) | \(1561\) |
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*l n(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*ar ctan(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*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)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+2*I*arctan(a*x)*polylog...
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=c \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx\right ) \]
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
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*arctan(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
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )}{x^3} \,d x \]