\(\int \frac {(a+b \arctan (c x))^2}{x^3 (d+e x)} \, dx\) [147]

Optimal result

Integrand size = 21, antiderivative size = 591 \[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+e x)} \, dx=-\frac {b c (a+b \arctan (c x))}{d x}-\frac {c^2 (a+b \arctan (c x))^2}{2 d}+\frac {i c e (a+b \arctan (c x))^2}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d x^2}+\frac {e (a+b \arctan (c x))^2}{d^2 x}+\frac {2 e^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d}+\frac {e^2 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d}-\frac {2 b c e (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {i b^2 c e \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b e^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b e^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^3}+\frac {b^2 e^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^3}-\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3} \]

[Out]

-b*c*(a+b*arctan(c*x))/d/x-1/2*c^2*(a+b*arctan(c*x))^2/d+I*b^2*c*e*polylog(2,-1+2/(1-I*c*x))/d^2-1/2*(a+b*arct
an(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*c*e*(a+b*arctan(c*x))^2/d^2+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*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+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] (verified)

Time = 0.56 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {4996, 4946, 5038, 272, 36, 29, 31, 5004, 5044, 4988, 2497, 4942, 5108, 5114, 6745, 4968} \[ \int \frac {(a+b \arctan (c x))^2}{x^3 (d+e x)} \, dx=\frac {2 e^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d^3}-\frac {c^2 (a+b \arctan (c x))^2}{2 d}-\frac {i b e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^3}-\frac {i b e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{d^3}+\frac {i b e^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d^3}+\frac {i b e^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {e^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{d^3}-\frac {e^2 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^3}+\frac {i c e (a+b \arctan (c x))^2}{d^2}+\frac {e (a+b \arctan (c x))^2}{d^2 x}-\frac {2 b c e \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}-\frac {(a+b \arctan (c x))^2}{2 d x^2}-\frac {b c (a+b \arctan (c x))}{d x}-\frac {b^2 c^2 \log \left (c^2 x^2+1\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 d^3}-\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 d^3}+\frac {b^2 e^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d^3}-\frac {b^2 e^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}+\frac {i b^2 c e \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{d^2} \]

[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 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 2497

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 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 4968

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] + S
imp[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] + Si
mp[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] &&
NeQ[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 4996

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

Mathematica [F(-1)]

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

[In]

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

[Out]

$Aborted

Maple [C] (warning: unable to verify)

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

Time = 53.89 (sec) , antiderivative size = 2804, normalized size of antiderivative = 4.74

method	result	size
parts	\(\text {Expression too large to display}\)	\(2804\)
derivativedivides	\(\text {Expression too large to display}\)	\(2853\)
default	\(\text {Expression too large to display}\)	\(2853\)

[In]

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

[Out]

-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*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2
/(c^2*x^2+1)+I*e+c*d))*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/((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*(-I*e*(1+I*c*x)^
2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/((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*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d))*csgn(I*(-I*e*(1+I*c*x)^2/(c^
2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/((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*P
i*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*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*c*e^2*arctan(c*x)^
2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(I*c*d+e)/d^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))^2*arcta
n(c*x)^2+a^2*(-1/2/d/x^2+e^2/d^3*ln(x)+e/d^2/x-e^2/d^3*ln(e*x+d))+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+I*b^2*e^3*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(I*c*d+e)-b^
2*c*e^2*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(I*c*d+e)/d^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+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)^2-1/2*I*b^2/d^3*e^2*Pi*csgn(I*(-I*e*(1+I*c*
x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/((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*c*e^2*polylo
g(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(I*c*d+e)/d^2-b^2*arctan(c*x)^2*e^2/d^3*ln(c*e*x+c*d)-I*b^2*c
^2*arctan(c*x)/d+I*b^2*c/d^2*e*arctan(c*x)^2-2*b^2*c/d^2*e*arctan(c*x)*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-b^2*e
^3*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(I*c*d+e)-2*I*b^2*c/d^2*e*dilog((1+I*c*
x)/(c^2*x^2+1)^(1/2))+2*I*b^2*c/d^2*e*dilog(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*b^2/d^3*e^2*Pi*arctan(c*x)^2-
2*I*b^2*e^2/d^3*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*b^2*e^2/d^3*arctan(c*x)*polylog(2,-(1+I
*c*x)/(c^2*x^2+1)^(1/2))+2*b^2*e^2/d^3*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*b^2*e^2/d^3*polylog(3,-(1+I*c*
x)/(c^2*x^2+1)^(1/2))-1/2*b^2*arctan(c*x)^2/d/x^2+b^2*c^2/d*ln((1+I*c*x)/(c^2*x^2+1)^(1/2)-1)+b^2*c^2/d*ln(1+(
1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*b^2*c^2/d*arctan(c*x)^2-b^2*c*arctan(c*x)/x/d+b^2*arctan(c*x)^2*e/d^2/x-1/2*b^
2*e^3*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/d^3/(I*c*d+e)-b^2*e^2/d^3*arctan(c*x)^2*ln((1+I*c
*x)^2/(c^2*x^2+1)-1)+b^2*e^2/d^3*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+b^2*e^2/d^3*arctan(c*x)^2*ln(
1+(1+I*c*x)/(c^2*x^2+1)^(1/2))+b^2*e^2/d^3*arctan(c*x)^2*ln(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*
x^2+1)+I*e+c*d)+b^2*arctan(c*x)^2*e^2/d^3*ln(c*x)+2*a*b*c^2*(-1/2*arctan(c*x)/d/c^2/x^2+1/c^2*arctan(c*x)*e^2/
d^3*ln(c*x)+1/c^2*arctan(c*x)*e/d^2/x-1/c^2*arctan(c*x)*e^2/d^3*ln(c*e*x+c*d)-1/2*c*(1/c^2/d^2*(-e*ln(c^2*x^2+
1)+d*c*arctan(c*x)+2*e*ln(c*x)+d/x)+2/d^3/c^3*e^2*(-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))-2/d^3/c^3*e^3*(-1/2*I*ln(c*e*x+c*d)*(ln((I*e-e*c*x)/(c*d+I*e))-ln((I*e+e*
c*x)/(I*e-c*d)))/e-1/2*I*(dilog((I*e-e*c*x)/(c*d+I*e))-dilog((I*e+e*c*x)/(I*e-c*d)))/e)))

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[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)