3.147 \(\int \frac {(a+b \tan ^{-1}(c x))^2}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=591 \[ -\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac {i b e^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}-\frac {i b e^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {i b e^2 \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {e^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^3}+\frac {2 e^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^3}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}-\frac {2 b c e \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 d^3}+\frac {b^2 e^2 \text {Li}_3\left (\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {i b^2 c e \text {Li}_2\left (\frac {2}{1-i c x}-1\right )}{d^2} \]

[Out]

-b*c*(a+b*arctan(c*x))/d/x-1/2*c^2*(a+b*arctan(c*x))^2/d+I*c*e*(a+b*arctan(c*x))^2/d^2-1/2*(a+b*arctan(c*x))^2
/d/x^2+e*(a+b*arctan(c*x))^2/d^2/x-2*e^2*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^3+b^2*c^2*ln(x)/d+e^2*(
a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/d^3-e^2*(a+b*arctan(c*x))^2*ln(2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/d^3-1/2*b^2
*c^2*ln(c^2*x^2+1)/d-2*b*c*e*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^2+I*b*e^2*(a+b*arctan(c*x))*polylog(2,-1+2/
(1+I*c*x))/d^3+I*b*e^2*(a+b*arctan(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/d^3-I*b*e^2*(a+b*arctan(
c*x))*polylog(2,1-2/(1+I*c*x))/d^3+I*b^2*c*e*polylog(2,-1+2/(1-I*c*x))/d^2-I*b*e^2*(a+b*arctan(c*x))*polylog(2
,1-2/(1-I*c*x))/d^3+1/2*b^2*e^2*polylog(3,1-2/(1-I*c*x))/d^3-1/2*b^2*e^2*polylog(3,1-2/(1+I*c*x))/d^3+1/2*b^2*
e^2*polylog(3,-1+2/(1+I*c*x))/d^3-1/2*b^2*e^2*polylog(3,1-2*c*(e*x+d)/(c*d+I*e)/(1-I*c*x))/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.84, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {4876, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610, 4858} \[ -\frac {i b e^2 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}-\frac {i b e^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {i b e^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^3}+\frac {b^2 e^2 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {b^2 e^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 d^3}+\frac {i b^2 c e \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {e^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^3}+\frac {2 e^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^3}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}-\frac {2 b c e \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 266

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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4850

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 4852

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

Rule 4858

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^2*Log[2/
(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] + Sim
p[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] - Simp[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 -
(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp
[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && Ne
Q[c^2*d^2 + e^2, 0]

Rule 4868

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 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4884

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 4918

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 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 6610

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*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^3 (d+e x)} \, dx &=\int \left (\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x^2}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{d^3 x}-\frac {e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx}{d}-\frac {e \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{d^3}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {(b c) \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx}{d}-\frac {(2 b c e) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {\left (4 b c e^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}\\ &=\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {(b c) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d}-\frac {\left (b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{d}-\frac {(2 i b c e) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx}{d^2}+\frac {\left (2 b c e^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {\left (2 b c e^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {2 b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (2 b^2 c^2 e\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (i b^2 c e^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {\left (i b^2 c e^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {2 b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b^2 c e \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {b^2 e^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {2 b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b^2 c e \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {b^2 e^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac {c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {i c e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac {2 e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d}-\frac {2 b c e \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b^2 c e \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {b^2 e^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {b^2 e^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}\\ \end {align*}

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Mathematica [A]  time = 21.88, size = 1173, normalized size = 1.98 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*a^2/(d*x^2) + (a^2*e)/(d^2*x) + (a^2*e^2*Log[x])/d^3 - (a^2*e^2*Log[d + e*x])/d^3 - (a*b*((c^2*d^3)/x + I
*c*d*e^2*Pi*ArcTan[c*x] - (2*c*d^2*e*ArcTan[c*x])/x + (c*d^3*(1 + c^2*x^2)*ArcTan[c*x])/x^2 - (2*I)*c*d*e^2*Ar
cTan[(c*d)/e]*ArcTan[c*x] + I*c*d*e^2*ArcTan[c*x]^2 + e^3*ArcTan[c*x]^2 - Sqrt[1 + (c^2*d^2)/e^2]*e^3*E^(I*Arc
Tan[(c*d)/e])*ArcTan[c*x]^2 + c*d*e^2*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*c*d*e^2*ArcTan[c*x]*Log[1 - E^((2
*I)*ArcTan[c*x])] + 2*c*d*e^2*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c*d*e^2*A
rcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c^2*d^2*e*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (c*
d*e^2*Pi*Log[1 + c^2*x^2])/2 - 2*c*d*e^2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*c*d*e^2*P
olyLog[2, E^((2*I)*ArcTan[c*x])] - I*c*d*e^2*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c*d^4) +
(b^2*((-I)*c*d*e^2*Pi^3 - (24*c^2*d^3*ArcTan[c*x])/x + (24*I)*c^2*d^2*e*ArcTan[c*x]^2 + (24*c*d^2*e*ArcTan[c*
x]^2)/x - (12*c*d^3*(1 + c^2*x^2)*ArcTan[c*x]^2)/x^2 - 16*e^3*ArcTan[c*x]^3 + 16*Sqrt[1 + (c^2*d^2)/e^2]*e^3*E
^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^3 + 24*c*d*e^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*c*d*e^2*Pi*
ArcTan[c*x]*Log[1 + E^((-2*I)*ArcTan[c*x])] - 48*c^2*d^2*e*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 48*c*d
*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 48*c*d*e^2*ArcTan[c*x]^2
*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 24*c*d*e^2*Pi*ArcTan[c*x]*Log[(-2*I)/(-I + c*x)] + 24*c^
3*d^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] - 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[(I + c*x + E^((2*I)*ArcTan[(c*
d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] + 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1
- E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] + 24*c*d*e^2
*ArcTan[c*x]^2*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan
[c*x]]] + 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (24*I)*c*d*e^2*ArcT
an[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*c^2*d^2*e*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (24*I)*c*d*e
^2*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*c*d*e^2*PolyLog[3, E^((-2*I)*ArcTan[
c*x])] - 12*c*d*e^2*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(24*c*d^4)

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{e x^{4} + d x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [C]  time = 65.50, size = 2861, normalized size = 4.84 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2/d/x^2+1/2*I*b^2/d^3*e^2*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-c*b^2*e^2*arctan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^2/(e+I*d*c)+I*a*b*e^2/d^3*l
n(c*x)*ln(1+I*c*x)+I*b^2*e^3*arctan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(e+I*d*c)+
I*a*b/d^3*e^2*ln(c*e*x+c*d)*ln((I*e+c*e*x)/(I*e-d*c))-1/2*I*c*b^2*e^2*polylog(3,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^
2/(c^2*x^2+1))/d^2/(e+I*d*c)-2*c*b^2/d^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)*e-2*c*a*b/d^2*e*ln(c*x)
+c*a*b/d^2*e*ln(c^2*x^2+1)+I*c*b^2/d^2*e*arctan(c*x)^2-I*a*b/d^3*e^2*dilog((I*e-c*e*x)/(d*c+I*e))+1/2*I*b^2/d^
3*e^2*Pi*arctan(c*x)^2-I*a*b*e^2/d^3*dilog(1-I*c*x)-2*I*b^2/d^3*e^2*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+
1)^(1/2))-2*I*b^2/d^3*e^2*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*c*b^2*dilog((1+I*c*x)/(c^2*x^
2+1)^(1/2))/d^2*e+2*I*c*b^2/d^2*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))*e-b^2*e^3*arctan(c*x)^2*ln(1-(I*e-d*c)/(d
*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(e+I*d*c)+I*a*b/d^3*e^2*dilog((I*e+c*e*x)/(I*e-d*c))+I*a*b*e^2/d^3*dilog(
1+I*c*x)+2*a*b*arctan(c*x)/d^2*e/x+2*a*b*arctan(c*x)/d^3*e^2*ln(c*x)-2*a*b*arctan(c*x)/d^3*e^2*ln(c*e*x+c*d)-1
/2*I*b^2/d^3*e^2*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))*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)^2-1/2*I*b^2/d^3*e^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*c
sgn(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+1/2*I*b^2/d^3*e^2*Pi*csgn(I*(-I
*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I
*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/2*I*b^2/d^3*e^2*Pi*csgn(I/((1+I*c*
x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c
^2*x^2+1)+1))^2*arctan(c*x)^2-I*c*b^2*e^2*arctan(c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^2/
(e+I*d*c)-1/2*I*b^2/d^3*e^2*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+1/2*I*b^2/d^3*e^2*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))*csgn(((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)^2-1/2*I*b^2/d^3*e^2*Pi*c
sgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*
csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(
c*x)^2+1/2*I*b^2/d^3*e^2*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-c*a*b/d/x-c^2*a*b/d*arctan(c*x)-c*b^2*arct
an(c*x)/x/d+b^2*arctan(c*x)^2/d^3*e^2*ln(c*x)-b^2*arctan(c*x)^2/d^3*e^2*ln(c*e*x+c*d)+b^2/d^3*e^2*arctan(c*x)^
2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+b^2*arctan(c*x)^2/d^2*e/x+b^2*e^2/d^3*arctan(c*x)^2*ln(-I*(1+I*c*x)^2/(c^2
*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)-b^2/d^3*e^2*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)-a*b*arc
tan(c*x)/d/x^2+b^2/d^3*e^2*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*b^2*e^3*polylog(3,(I*e-d*c)/(d*
c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(e+I*d*c)-I*c^2*b^2*arctan(c*x)/d-1/2*c^2*b^2/d*arctan(c*x)^2+c^2*b^2/d*ln
((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)+c^2*b^2/d*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*b^2/d^3*e^2*polylog(3,-(1+I*c*x)
/(c^2*x^2+1)^(1/2))+2*b^2/d^3*e^2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+a^2/d^3*e^2*ln(c*x)-a^2/d^3*e^2*ln(c*
e*x+c*d)-1/2*b^2*arctan(c*x)^2/d/x^2+a^2/d^2*e/x-1/2*I*b^2/d^3*e^2*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d
*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-1/2*I*b^2/d^3*e^2*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+1/2*I*b^2/d^3*e^2*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*arctan(c*x)^2-I*a*b/d^3*e^2*ln(c*e*x+c*d)*ln((I*e-c*e*x)/(d*c+I*e
))-I*a*b*e^2/d^3*ln(c*x)*ln(1-I*c*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a^{2} {\left (\frac {2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \relax (x)}{d^{3}} - \frac {2 \, e x - d}{d^{2} x^{2}}\right )} + \frac {2 \, d^{2} x^{2} \int \frac {12 \, {\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + {\left (b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 4 \, {\left (2 \, b^{2} c e^{2} x^{3} - b^{2} c d^{2} x - 8 \, a b d^{2} - {\left (8 \, a b c^{2} d^{2} - b^{2} c d e\right )} x^{2}\right )} \arctan \left (c x\right ) + 2 \, {\left (2 \, b^{2} c^{2} e^{2} x^{4} + b^{2} c^{2} d e x^{3} - b^{2} c^{2} d^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2} d^{2} e x^{6} + c^{2} d^{3} x^{5} + d^{2} e x^{4} + d^{3} x^{3}}\,{d x} + 4 \, {\left (2 \, b^{2} e x - b^{2} d\right )} \arctan \left (c x\right )^{2} - {\left (2 \, b^{2} e x - b^{2} d\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \, d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - d)/(d^2*x^2)) + 1/32*(32*d^2*x^2*integrate(1/16
*(12*(b^2*c^2*d^2*x^2 + b^2*d^2)*arctan(c*x)^2 + (b^2*c^2*d^2*x^2 + b^2*d^2)*log(c^2*x^2 + 1)^2 - 4*(2*b^2*c*e
^2*x^3 - b^2*c*d^2*x - 8*a*b*d^2 - (8*a*b*c^2*d^2 - b^2*c*d*e)*x^2)*arctan(c*x) + 2*(2*b^2*c^2*e^2*x^4 + b^2*c
^2*d*e*x^3 - b^2*c^2*d^2*x^2)*log(c^2*x^2 + 1))/(c^2*d^2*e*x^6 + c^2*d^3*x^5 + d^2*e*x^4 + d^3*x^3), x) + 4*(2
*b^2*e*x - b^2*d)*arctan(c*x)^2 - (2*b^2*e*x - b^2*d)*log(c^2*x^2 + 1)^2)/(d^2*x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^3\,\left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{x^{3} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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