3.13.0 ∫ (d+ex2)2(a+b arctan(cx))2 _______________________________________

Integrand size = 23, antiderivative size = 320 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=-\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^2 \log (x)-\frac {1}{2} b^2 c^2 d^2 \log \left (1+c^2 x^2\right )+\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{2 c^2}-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+b^2 d e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]

-a*b*e^2*x/c-b^2*e^2*x*arctan(c*x)/c-b*c*d^2*(a+b*arctan(c*x))/x-1/2*c^2*d^2*(a+b*arctan(c*x))^2+1/2*e^2*(a+b*

arctan(c*x))^2/c^2-1/2*d^2*(a+b*arctan(c*x))^2/x^2+1/2*e^2*x^2*(a+b*arctan(c*x))^2-4*d*e*(a+b*arctan(c*x))^2*a

rctanh(-1+2/(1+I*c*x))+b^2*c^2*d^2*ln(x)-1/2*b^2*c^2*d^2*ln(c^2*x^2+1)+1/2*b^2*e^2*ln(c^2*x^2+1)/c^2-2*I*b*d*e

*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))+2*I*b*d*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-b^2*d*e*poly

log(3,1-2/(1+I*c*x))+b^2*d*e*polylog(3,-1+2/(1+I*c*x))

Rubi [A] (veriﬁed)

Time = 0.43 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5100, 4946, 5038, 272, 36, 29, 31, 5004, 4942, 5108, 5114, 6745, 5036, 4930, 266} \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=4 d e \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}-\frac {b c d^2 (a+b \arctan (c x))}{x}-2 i b d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+2 i b d e \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2-\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {1}{2} b^2 c^2 d^2 \log \left (c^2 x^2+1\right )+b^2 c^2 d^2 \log (x)+\frac {b^2 e^2 \log \left (c^2 x^2+1\right )}{2 c^2}-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+b^2 d e \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) \]

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

-((a*b*e^2*x)/c) - (b^2*e^2*x*ArcTan[c*x])/c - (b*c*d^2*(a + b*ArcTan[c*x]))/x - (c^2*d^2*(a + b*ArcTan[c*x])^

2)/2 + (e^2*(a + b*ArcTan[c*x])^2)/(2*c^2) - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) + (e^2*x^2*(a + b*ArcTan[c*x]

)^2)/2 + 4*d*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^2*Log[x] - (b^2*c^2*d^2*Log[1 + c^

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

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

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

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

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]

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ

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]]

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

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

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]

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]

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/

e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), 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] && GtQ[m, 1]

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]

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With

[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,

c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

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)))^

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

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 (a+b \arctan (c x))^2}{x^3}+\frac {2 d e (a+b \arctan (c x))^2}{x}+e^2 x (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int \frac {(a+b \arctan (c x))^2}{x^3} \, dx+(2 d e) \int \frac {(a+b \arctan (c x))^2}{x} \, dx+e^2 \int x (a+b \arctan (c x))^2 \, dx \\ & = -\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+\left (b c d^2\right ) \int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx-(8 b c d e) \int \frac {(a+b \arctan (c x)) \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (b c e^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+\left (b c d^2\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx-\left (b c^3 d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx+(4 b c d e) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-(4 b c d e) \int \frac {(a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {\left (b e^2\right ) \int (a+b \arctan (c x)) \, dx}{c}+\frac {\left (b e^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c} \\ & = -\frac {a b e^2 x}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\left (b^2 c^2 d^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx+\left (2 i b^2 c d e\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 i b^2 c d e\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {\left (b^2 e^2\right ) \int \arctan (c x) \, dx}{c} \\ & = -\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+b^2 d e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {1}{2} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (b^2 e^2\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{2 c^2}-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+b^2 d e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )+\frac {1}{2} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b^2 c^4 d^2\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a b e^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{c}-\frac {b c d^2 (a+b \arctan (c x))}{x}-\frac {1}{2} c^2 d^2 (a+b \arctan (c x))^2+\frac {e^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d^2 (a+b \arctan (c x))^2}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))^2+4 d e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^2 \log (x)-\frac {1}{2} b^2 c^2 d^2 \log \left (1+c^2 x^2\right )+\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{2 c^2}-2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+2 i b d e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-b^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+b^2 d e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.41 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^3} \, dx=\frac {1}{2} \left (-\frac {a^2 d^2}{x^2}+a^2 e^2 x^2+\frac {2 a b e^2 \left (-c x+\left (1+c^2 x^2\right ) \arctan (c x)\right )}{c^2}-\frac {2 a b d^2 (\arctan (c x)+c x (1+c x \arctan (c x)))}{x^2}+4 a^2 d e \log (x)-\frac {b^2 d^2 \left (2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )}{x^2}+\frac {b^2 e^2 \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )}{c^2}+4 i a b d e (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+\frac {1}{6} b^2 d e \left (-i \pi ^3+16 i \arctan (c x)^3+24 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-24 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+24 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )\right ) \]

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

(-((a^2*d^2)/x^2) + a^2*e^2*x^2 + (2*a*b*e^2*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]))/c^2 - (2*a*b*d^2*(ArcTan[c*

x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 + 4*a^2*d*e*Log[x] - (b^2*d^2*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c

*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]))/x^2 + (b^2*e^2*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x

]^2 + Log[1 + c^2*x^2]))/c^2 + (4*I)*a*b*d*e*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*d*e*((-I)*Pi^3

+ (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]^2*Log[1 + E^((2*I)*

ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I

)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/6)/2

Maple [C] (warning: unable to verify)

Time = 15.75 (sec) , antiderivative size = 1500, normalized size of antiderivative = 4.69


method	result	size

parts	\(\text {Expression too large to display}\)	\(1500\)
derivativedivides	\(\text {Expression too large to display}\)	\(1521\)
default	\(\text {Expression too large to display}\)	\(1521\)

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

I*b^2*d*e*Pi*arctan(c*x)^2+2*I*b^2*d*e*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-4*I*b^2*polylog(2,-(1+I

*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)*d*e-4*I*b^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)*d*e-b^2*e^

2*x*arctan(c*x)/c+2*a*b*c^2*(1/2*arctan(c*x)/c^2*e^2*x^2+2*arctan(c*x)/c^2*d*e*ln(c*x)-1/2*arctan(c*x)*d^2/c^2

/x^2-1/2/c^4*(c*x*e^2+c^3*d^2/x+(c^4*d^2-e^2)*arctan(c*x)+4*c^2*d*e*(-1/2*I*ln(c*x)*ln(1+I*c*x)+1/2*I*ln(c*x)*

ln(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*x))))+I*b^2*d*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn

(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+

4*b^2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))*d*e-b^2*d*e*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+4*b^2*polylog(3,-

(1+I*c*x)/(c^2*x^2+1)^(1/2))*d*e+1/2*b^2*arctan(c*x)^2*e^2*x^2-1/2*b^2*arctan(c*x)^2*d^2/x^2-b^2/c^2*e^2*ln((1

+I*c*x)^2/(c^2*x^2+1)+1)+1/2*b^2/c^2*e^2*arctan(c*x)^2+b^2*c^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))*d^2+b^2*c^2*l

n((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)*d^2-1/2*b^2*c^2*arctan(c*x)^2*d^2-I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-

1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^

2*x^2+1)+1))^3*arctan(c*x)^2+I*b^2*d*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*ar

ctan(c*x)^2+I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^

2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-I*b^2*d*e*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I

*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-I*b^2*d*e*Pi*csgn(((1+I*c*x)^2/(c^2*

x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arcta

n(c*x)^2-I*b^2*d*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2

*x^2+1)+1))^2*arctan(c*x)^2+a^2*(1/2*e^2*x^2+2*e*d*ln(x)-1/2*d^2/x^2)+2*b^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))*

arctan(c*x)^2*d*e+2*b^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)^2*d*e+I*b^2/c^2*arctan(c*x)*e^2-b^2*c*d^

2*arctan(c*x)/x+2*b^2*arctan(c*x)^2*d*e*ln(c*x)-2*b^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)*arctan(c*x)^2*d*e-I*b^2*c^

Fricas [F]

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

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

integral((a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*arctan(c*x)^2 + 2*(a

*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*arctan(c*x))/x^3, x)

Sympy [F]

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

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

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

Maxima [F]

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

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

1/2*a^2*e^2*x^2 - ((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*a*b*d^2 + 2*a^2*d*e*log(x) - 1/2*a^2*d^2/x^2 + 1

/96*((1152*b^2*c^2*e^2*integrate(1/16*x^6*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*c^2*e^2*integrate(1/16*x^

6*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 3072*a*b*c^2*e^2*integrate(1/16*x^6*arctan(c*x)/(c^2*x^5 + x^3), x)

+ 192*b^2*c^2*e^2*integrate(1/16*x^6*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) + 2304*b^2*c^2*d*e*integrate(1/16*x

^4*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 6144*a*b*c^2*d*e*integrate(1/16*x^4*arctan(c*x)/(c^2*x^5 + x^3), x) + 1

152*b^2*c^2*d^2*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^5 + x^3), x) + 96*b^2*c^2*d^2*integrate(1/16*x^2*log(c

^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) - 192*b^2*c^2*d^2*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^5 + x^3), x) +

2*b^2*d*e*log(c^2*x^2 + 1)^3 - 384*b^2*c*e^2*integrate(1/16*x^5*arctan(c*x)/(c^2*x^5 + x^3), x) + 384*b^2*c*d^

2*integrate(1/16*x*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*e^2*integrate(1/16*x^4*arctan(c*x)^2/(c^2*x^5 +

x^3), x) + 3072*a*b*e^2*integrate(1/16*x^4*arctan(c*x)/(c^2*x^5 + x^3), x) + 2304*b^2*d*e*integrate(1/16*x^2*a

rctan(c*x)^2/(c^2*x^5 + x^3), x) + 192*b^2*d*e*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + 614

4*a*b*d*e*integrate(1/16*x^2*arctan(c*x)/(c^2*x^5 + x^3), x) + 1152*b^2*d^2*integrate(1/16*arctan(c*x)^2/(c^2*

x^5 + x^3), x) + 96*b^2*d^2*integrate(1/16*log(c^2*x^2 + 1)^2/(c^2*x^5 + x^3), x) + b^2*e^2*log(c^2*x^2 + 1)^3

/c^2)*x^2 + 12*(b^2*e^2*x^4 - b^2*d^2)*arctan(c*x)^2 - 3*(b^2*e^2*x^4 - b^2*d^2)*log(c^2*x^2 + 1)^2)/x^2

Giac [F(-1)]

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

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

Mupad [F(-1)]

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

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

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

\(\int \frac {(d+e x^2)^2 (a+b \arctan (c x))^2}{x^3} \, dx\) [1260]

Optimal result