\(\int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx\) [102]

Optimal result

Integrand size = 25, antiderivative size = 365 \[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=-\frac {b^2 c^2}{3 d x}-\frac {b^2 c^3 \arctan (c x)}{3 d}-\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i b c^2 (a+b \arctan (c x))}{d x}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}-\frac {i b^2 c^3 \log (x)}{d}+\frac {i b^2 c^3 \log \left (1+c^2 x^2\right )}{2 d}-\frac {8 b c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {4 i b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{3 d}-\frac {b c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d} \]

[Out]

-1/3*b^2*c^2/d/x-1/3*b^2*c^3*arctan(c*x)/d-1/3*b*c*(a+b*arctan(c*x))/d/x^2+I*b*c^2*(a+b*arctan(c*x))/d/x+11/6*
I*c^3*(a+b*arctan(c*x))^2/d-1/3*(a+b*arctan(c*x))^2/d/x^3+1/2*I*c*(a+b*arctan(c*x))^2/d/x^2+c^2*(a+b*arctan(c*
x))^2/d/x-I*b^2*c^3*ln(x)/d+1/2*I*b^2*c^3*ln(c^2*x^2+1)/d-8/3*b*c^3*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d+I*c^
3*(a+b*arctan(c*x))^2*ln(2-2/(1+I*c*x))/d+4/3*I*b^2*c^3*polylog(2,-1+2/(1-I*c*x))/d-b*c^3*(a+b*arctan(c*x))*po
lylog(2,-1+2/(1+I*c*x))/d+1/2*I*b^2*c^3*polylog(3,-1+2/(1+I*c*x))/d

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4990, 4946, 5038, 331, 209, 5044, 4988, 2497, 272, 36, 29, 31, 5004, 5114, 6745} \[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=-\frac {b c^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{d}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {8 b c^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{3 d}+\frac {i c^3 \log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{d}+\frac {c^2 (a+b \arctan (c x))^2}{d x}+\frac {i b c^2 (a+b \arctan (c x))}{d x}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}-\frac {b c (a+b \arctan (c x))}{3 d x^2}-\frac {b^2 c^3 \arctan (c x)}{3 d}+\frac {4 i b^2 c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )}{3 d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{2 d}-\frac {i b^2 c^3 \log (x)}{d}-\frac {b^2 c^2}{3 d x}+\frac {i b^2 c^3 \log \left (c^2 x^2+1\right )}{2 d} \]

[In]

Int[(a + b*ArcTan[c*x])^2/(x^4*(d + I*c*d*x)),x]

[Out]

-1/3*(b^2*c^2)/(d*x) - (b^2*c^3*ArcTan[c*x])/(3*d) - (b*c*(a + b*ArcTan[c*x]))/(3*d*x^2) + (I*b*c^2*(a + b*Arc
Tan[c*x]))/(d*x) + (((11*I)/6)*c^3*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(3*d*x^3) + ((I/2)*c*(a +
b*ArcTan[c*x])^2)/(d*x^2) + (c^2*(a + b*ArcTan[c*x])^2)/(d*x) - (I*b^2*c^3*Log[x])/d + ((I/2)*b^2*c^3*Log[1 +
c^2*x^2])/d - (8*b*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/(3*d) + (I*c^3*(a + b*ArcTan[c*x])^2*Log[2
- 2/(1 + I*c*x)])/d + (((4*I)/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (b*c^3*(a + b*ArcTan[c*x])*PolyLo
g[2, -1 + 2/(1 + I*c*x)])/d + ((I/2)*b^2*c^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/d

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 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 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 4990

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[1/d, I
nt[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f), Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^p/(d + e*x)),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[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 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 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}& = -\left ((i c) \int \frac {(a+b \arctan (c x))^2}{x^3 (d+i c d x)} \, dx\right )+\frac {\int \frac {(a+b \arctan (c x))^2}{x^4} \, dx}{d} \\ & = -\frac {(a+b \arctan (c x))^2}{3 d x^3}-c^2 \int \frac {(a+b \arctan (c x))^2}{x^2 (d+i c d x)} \, dx-\frac {(i c) \int \frac {(a+b \arctan (c x))^2}{x^3} \, dx}{d}+\frac {(2 b c) \int \frac {a+b \arctan (c x)}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d} \\ & = -\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\left (i c^3\right ) \int \frac {(a+b \arctan (c x))^2}{x (d+i c d x)} \, dx+\frac {(2 b c) \int \frac {a+b \arctan (c x)}{x^3} \, dx}{3 d}-\frac {c^2 \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx}{d}-\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx}{d}-\frac {\left (2 b c^3\right ) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d} \\ & = -\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i c^3 (a+b \arctan (c x))^2}{3 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {\left (i b c^2\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{3 d}-\frac {\left (2 i b c^3\right ) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{3 d}-\frac {\left (2 b c^3\right ) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (i b c^4\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{d}-\frac {\left (2 i b c^4\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} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i b c^2 (a+b \arctan (c x))}{d x}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}-\frac {2 b c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {b c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}-\frac {\left (2 i b c^3\right ) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx}{d}-\frac {\left (i b^2 c^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {\left (b^2 c^4\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 d}+\frac {\left (2 b^2 c^4\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{3 d}+\frac {\left (b^2 c^4\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b^2 c^3 \arctan (c x)}{3 d}-\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i b c^2 (a+b \arctan (c x))}{d x}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}-\frac {8 b c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{3 d}-\frac {b c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac {\left (2 b^2 c^4\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b^2 c^3 \arctan (c x)}{3 d}-\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i b c^2 (a+b \arctan (c x))}{d x}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}-\frac {8 b c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {4 i b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{3 d}-\frac {b c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (i b^2 c^5\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {b^2 c^2}{3 d x}-\frac {b^2 c^3 \arctan (c x)}{3 d}-\frac {b c (a+b \arctan (c x))}{3 d x^2}+\frac {i b c^2 (a+b \arctan (c x))}{d x}+\frac {11 i c^3 (a+b \arctan (c x))^2}{6 d}-\frac {(a+b \arctan (c x))^2}{3 d x^3}+\frac {i c (a+b \arctan (c x))^2}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))^2}{d x}-\frac {i b^2 c^3 \log (x)}{d}+\frac {i b^2 c^3 \log \left (1+c^2 x^2\right )}{2 d}-\frac {8 b c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x))^2 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {4 i b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{3 d}-\frac {b c^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=-\frac {a^2}{3 d x^3}+\frac {i a^2 c}{2 d x^2}+\frac {a^2 c^2}{d x}+\frac {a^2 c^3 \arctan (c x)}{d}+\frac {i a^2 c^3 \log (x)}{d}-\frac {i a^2 c^3 \log \left (1+c^2 x^2\right )}{2 d}-\frac {2 i a b c^3 \left (-\frac {1}{2 c x}-\frac {i \left (1+c^2 x^2\right )}{6 c^2 x^2}+\frac {4 i \arctan (c x)}{3 c x}-\frac {i \left (1+c^2 x^2\right ) \arctan (c x)}{3 c^3 x^3}-\frac {\left (1+c^2 x^2\right ) \arctan (c x)}{2 c^2 x^2}+\frac {1}{2} i \arctan (c x)^2-\arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-\frac {4}{3} i \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+\frac {1}{2} i \left (\arctan (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )\right )}{d}+\frac {b^2 c^3 \left (\pi ^3-\frac {8}{c x}+\frac {24 i \arctan (c x)}{c x}-\frac {8 \left (1+c^2 x^2\right ) \arctan (c x)}{c^2 x^2}+32 i \arctan (c x)^2+\frac {32 \arctan (c x)^2}{c x}-\frac {8 \left (1+c^2 x^2\right ) \arctan (c x)^2}{c^3 x^3}+\frac {12 i \left (1+c^2 x^2\right ) \arctan (c x)^2}{c^2 x^2}+24 i \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-64 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )-24 i \log (c x)-24 i \log \left (\frac {1}{\sqrt {1+c^2 x^2}}\right )-24 \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+32 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )\right )}{24 d} \]

[In]

Integrate[(a + b*ArcTan[c*x])^2/(x^4*(d + I*c*d*x)),x]

[Out]

-1/3*a^2/(d*x^3) + ((I/2)*a^2*c)/(d*x^2) + (a^2*c^2)/(d*x) + (a^2*c^3*ArcTan[c*x])/d + (I*a^2*c^3*Log[x])/d -
((I/2)*a^2*c^3*Log[1 + c^2*x^2])/d - ((2*I)*a*b*c^3*(-1/2*1/(c*x) - ((I/6)*(1 + c^2*x^2))/(c^2*x^2) + (((4*I)/
3)*ArcTan[c*x])/(c*x) - ((I/3)*(1 + c^2*x^2)*ArcTan[c*x])/(c^3*x^3) - ((1 + c^2*x^2)*ArcTan[c*x])/(2*c^2*x^2)
+ (I/2)*ArcTan[c*x]^2 - ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - ((4*I)/3)*Log[(c*x)/Sqrt[1 + c^2*x^2]] +
(I/2)*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])))/d + (b^2*c^3*(Pi^3 - 8/(c*x) + ((24*I)*ArcTan[c*x]
)/(c*x) - (8*(1 + c^2*x^2)*ArcTan[c*x])/(c^2*x^2) + (32*I)*ArcTan[c*x]^2 + (32*ArcTan[c*x]^2)/(c*x) - (8*(1 +
c^2*x^2)*ArcTan[c*x]^2)/(c^3*x^3) + ((12*I)*(1 + c^2*x^2)*ArcTan[c*x]^2)/(c^2*x^2) + (24*I)*ArcTan[c*x]^2*Log[
1 - E^((-2*I)*ArcTan[c*x])] - 64*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - (24*I)*Log[c*x] - (24*I)*Log[1/S
qrt[1 + c^2*x^2]] - 24*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (32*I)*PolyLog[2, E^((2*I)*ArcTan[c*x]
)] + (12*I)*PolyLog[3, E^((-2*I)*ArcTan[c*x])]))/(24*d)

Maple [F(-1)]

Timed out.

hanged

[In]

int((a+b*arctan(c*x))^2/x^4/(d+I*c*d*x),x)

[Out]

int((a+b*arctan(c*x))^2/x^4/(d+I*c*d*x),x)

Fricas [F]

\[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )} x^{4}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))^2/x^4/(d+I*c*d*x),x, algorithm="fricas")

[Out]

1/24*(-6*I*b^2*c^3*x^3*log(2*c*x/(c*x - I))*log(-(c*x + I)/(c*x - I))^2 - 12*I*b^2*c^3*x^3*dilog(-2*c*x/(c*x -
I) + 1)*log(-(c*x + I)/(c*x - I)) + 12*I*b^2*c^3*x^3*polylog(3, -(c*x + I)/(c*x - I)) + 24*d*x^3*integral(1/6
*(-6*I*a^2*c*x + 6*a^2 + (-6*I*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + 2*(3*a*b + I*b^2)*c*x + 6*I*a*b)*log(-(c*x + I)/(
c*x - I)))/(c^2*d*x^6 + d*x^4), x) - (6*b^2*c^2*x^2 + 3*I*b^2*c*x - 2*b^2)*log(-(c*x + I)/(c*x - I))^2)/(d*x^3
)

Sympy [F]

\[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=- \frac {i \left (\int \frac {a^{2}}{c x^{5} - i x^{4}}\, dx + \int \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c x^{5} - i x^{4}}\, dx + \int \frac {2 a b \operatorname {atan}{\left (c x \right )}}{c x^{5} - i x^{4}}\, dx\right )}{d} \]

[In]

integrate((a+b*atan(c*x))**2/x**4/(d+I*c*d*x),x)

[Out]

-I*(Integral(a**2/(c*x**5 - I*x**4), x) + Integral(b**2*atan(c*x)**2/(c*x**5 - I*x**4), x) + Integral(2*a*b*at
an(c*x)/(c*x**5 - I*x**4), x))/d

Maxima [F]

\[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )} x^{4}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))^2/x^4/(d+I*c*d*x),x, algorithm="maxima")

[Out]

-1/6*(6*I*c^3*log(I*c*x + 1)/d - 6*I*c^3*log(x)/d - (6*c^2*x^2 + 3*I*c*x - 2)/(d*x^3))*a^2 + 1/96*(24*b^2*c^3*
x^3*arctan(c*x)^3 + 3*I*b^2*c^3*x^3*log(c^2*x^2 + 1)^3 - 2*I*(1152*b^2*c^5*integrate(1/48*x^5*arctan(c*x)^2/(c
^2*d*x^6 + d*x^4), x) + b^2*c^3*log(c^2*x^2 + 1)^3/d + 12*b^2*c^3*arctan(c*x)^2/d + 288*b^2*c^3*integrate(1/48
*x^3*log(c^2*x^2 + 1)/(c^2*d*x^6 + d*x^4), x) + 192*b^2*c^2*integrate(1/48*x^2*arctan(c*x)/(c^2*d*x^6 + d*x^4)
, x) + 1728*b^2*c*integrate(1/48*x*arctan(c*x)^2/(c^2*d*x^6 + d*x^4), x) + 144*b^2*c*integrate(1/48*x*log(c^2*
x^2 + 1)^2/(c^2*d*x^6 + d*x^4), x) + 4608*a*b*c*integrate(1/48*x*arctan(c*x)/(c^2*d*x^6 + d*x^4), x) - 192*b^2
*c*integrate(1/48*x*log(c^2*x^2 + 1)/(c^2*d*x^6 + d*x^4), x) + 576*b^2*integrate(1/48*arctan(c*x)*log(c^2*x^2
+ 1)/(c^2*d*x^6 + d*x^4), x))*d*x^3 - 16*(b^2*c^3*arctan(c*x)^3/d + 36*b^2*c^4*integrate(1/48*x^4*log(c^2*x^2
+ 1)^2/(c^2*d*x^6 + d*x^4), x) - 72*b^2*c^4*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*d*x^6 + d*x^4), x) + 72*b
^2*c^3*integrate(1/48*x^3*arctan(c*x)/(c^2*d*x^6 + d*x^4), x) - 12*b^2*c^2*integrate(1/48*x^2*log(c^2*x^2 + 1)
/(c^2*d*x^6 + d*x^4), x) + 72*b^2*c*integrate(1/48*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*d*x^6 + d*x^4), x) - 48
*b^2*c*integrate(1/48*x*arctan(c*x)/(c^2*d*x^6 + d*x^4), x) - 216*b^2*integrate(1/48*arctan(c*x)^2/(c^2*d*x^6
+ d*x^4), x) - 18*b^2*integrate(1/48*log(c^2*x^2 + 1)^2/(c^2*d*x^6 + d*x^4), x) - 576*a*b*integrate(1/48*arcta
n(c*x)/(c^2*d*x^6 + d*x^4), x))*d*x^3 + 4*(6*b^2*c^2*x^2 + 3*I*b^2*c*x - 2*b^2)*arctan(c*x)^2 + (6*b^2*c^3*x^3
*arctan(c*x) - 6*b^2*c^2*x^2 - 3*I*b^2*c*x + 2*b^2)*log(c^2*x^2 + 1)^2 + 4*(3*I*b^2*c^3*x^3*arctan(c*x)^2 + (6
*I*b^2*c^2*x^2 - 3*b^2*c*x - 2*I*b^2)*arctan(c*x))*log(c^2*x^2 + 1))/(d*x^3)

Giac [F]

\[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )} x^{4}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))^2/x^4/(d+I*c*d*x),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^4 (d+i c d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^4\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]

[In]

int((a + b*atan(c*x))^2/(x^4*(d + c*d*x*1i)),x)

[Out]

int((a + b*atan(c*x))^2/(x^4*(d + c*d*x*1i)), x)