\(\int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx\) [80]

Optimal result

Integrand size = 25, antiderivative size = 300 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {5}{2} d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \log \left (1+c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]

[Out]

a*b*c*d^2*x+b^2*c*d^2*x*arctan(c*x)-5/2*d^2*(a+b*arctan(c*x))^2+2*I*c*d^2*x*(a+b*arctan(c*x))^2-1/2*c^2*d^2*x^
2*(a+b*arctan(c*x))^2-2*d^2*(a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))+4*I*b*d^2*(a+b*arctan(c*x))*ln(2/(1+I*
c*x))-1/2*b^2*d^2*ln(c^2*x^2+1)-2*b^2*d^2*polylog(2,1-2/(1+I*c*x))-I*b*d^2*(a+b*arctan(c*x))*polylog(2,1-2/(1+
I*c*x))+I*b*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d^2*polylog(3,1-2/(1+I*c*x))+1/2*b^2*d^2*p
olylog(3,-1+2/(1+I*c*x))

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {4996, 4930, 5040, 4964, 2449, 2352, 4942, 5108, 5004, 5114, 6745, 4946, 5036, 266} \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=2 d^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2-i b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+i b d^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))+2 i c d^2 x (a+b \arctan (c x))^2-\frac {5}{2} d^2 (a+b \arctan (c x))^2+4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))+a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {1}{2} b^2 d^2 \log \left (c^2 x^2+1\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) \]

[In]

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

[Out]

a*b*c*d^2*x + b^2*c*d^2*x*ArcTan[c*x] - (5*d^2*(a + b*ArcTan[c*x])^2)/2 + (2*I)*c*d^2*x*(a + b*ArcTan[c*x])^2
- (c^2*d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (4*I)*b*d^2
*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*d^2*Log[1 + c^2*x^2])/2 - 2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x
)] - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2
/(1 + I*c*x)] - (b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2

Rule 266

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4930

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
] && (EqQ[n, 1] || EqQ[p, 1])

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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 5036

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]

Rule 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[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 (2 i c d^2 (a+b \arctan (c x))^2+\frac {d^2 (a+b \arctan (c x))^2}{x}-c^2 d^2 x (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int \frac {(a+b \arctan (c x))^2}{x} \, dx+\left (2 i c d^2\right ) \int (a+b \arctan (c x))^2 \, dx-\left (c^2 d^2\right ) \int x (a+b \arctan (c x))^2 \, dx \\ & = 2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-\left (4 b c d^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-\left (4 i b c^2 d^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\left (b c^3 d^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -2 d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+\left (4 i b c d^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx+\left (b c d^2\right ) \int (a+b \arctan (c x)) \, dx-\left (b c d^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx+\left (2 b c d^2\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b c d^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 \\ & = a b c d^2 x-\frac {5}{2} d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\left (i b^2 c d^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (i b^2 c d^2\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (4 i b^2 c d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (b^2 c d^2\right ) \int \arctan (c x) \, dx \\ & = a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {5}{2} d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )-\left (4 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )-\left (b^2 c^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = a b c d^2 x+b^2 c d^2 x \arctan (c x)-\frac {5}{2} d^2 (a+b \arctan (c x))^2+2 i c d^2 x (a+b \arctan (c x))^2-\frac {1}{2} c^2 d^2 x^2 (a+b \arctan (c x))^2+2 d^2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+4 i b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \log \left (1+c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b d^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.20 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))^2}{x} \, dx=\frac {1}{2} d^2 \left (4 i a^2 c x-a^2 c^2 x^2+2 b^2 c x \arctan (c x)-b^2 \left (1+c^2 x^2\right ) \arctan (c x)^2-2 a b \left (-c x+\left (1+c^2 x^2\right ) \arctan (c x)\right )+2 a^2 \log (c x)+4 i a b \left (2 c x \arctan (c x)-\log \left (1+c^2 x^2\right )\right )-b^2 \log \left (1+c^2 x^2\right )+4 b^2 \left (\arctan (c x) \left ((1+i c x) \arctan (c x)+2 i \log \left (1+e^{2 i \arctan (c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+2 i a b (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+2 b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} i \arctan (c x)^3+\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )\right ) \]

[In]

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

[Out]

(d^2*((4*I)*a^2*c*x - a^2*c^2*x^2 + 2*b^2*c*x*ArcTan[c*x] - b^2*(1 + c^2*x^2)*ArcTan[c*x]^2 - 2*a*b*(-(c*x) +
(1 + c^2*x^2)*ArcTan[c*x]) + 2*a^2*Log[c*x] + (4*I)*a*b*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) - b^2*Log[1 + c
^2*x^2] + 4*b^2*(ArcTan[c*x]*((1 + I*c*x)*ArcTan[c*x] + (2*I)*Log[1 + E^((2*I)*ArcTan[c*x])]) + PolyLog[2, -E^
((2*I)*ArcTan[c*x])]) + (2*I)*a*b*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 2*b^2*((-1/24*I)*Pi^3 + ((2*I)/
3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])
] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + Poly
Log[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)))/2

Maple [C] (warning: unable to verify)

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

Time = 29.47 (sec) , antiderivative size = 1317, normalized size of antiderivative = 4.39

method	result	size
parts	\(\text {Expression too large to display}\)	\(1317\)
derivativedivides	\(\text {Expression too large to display}\)	\(1319\)
default	\(\text {Expression too large to display}\)	\(1319\)

[In]

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

[Out]

a^2*d^2*(-1/2*c^2*x^2+2*I*c*x+ln(x))+b^2*d^2*(-1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+2*polylog(3,(1+I*c*x)/(
c^2*x^2+1)^(1/2))+2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*c^2*x^2*arctan(c*x)^2+ln(1+(1+I*c*x)^2/(c^2*x^
2+1))+3/2*arctan(c*x)^2+I*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+4*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1
/2))+4*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)*(c*x-I)-arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+
arctan(c*x)^2*ln(c*x)+arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)
^(1/2))-2*I*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1
)^(1/2))+1/2*I*Pi*arctan(c*x)^2+1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+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+(1+I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2+1/2*I*Pi*csgn(I*((1+I*c*x)^2/
(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*arct
an(c*x)^2-1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*
x^2+1)))^2*arctan(c*x)^2-1/2*I*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+(1
+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+2*I*arctan(c*x)^2*c*x+1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+
I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2-1/2*I*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^
2*arctan(c*x)^2+1/2*I*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*arctan(c*x)^2-1/2*I
*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/(1+(1+I*c
*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+4*I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+4*I*arctan(c*x)*ln(1-I
*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+2*a*d^2*b*(-1/2*c^2*x^2*arctan(c*x)+2*I*arctan(c*x)*c*x+arctan(c*x)*ln(c*x)+1/2
*c*x-I*ln(c^2*x^2+1)-1/2*arctan(c*x)+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))

Fricas [F]

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

[In]

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

[Out]

integral(-1/4*(4*a^2*c^2*d^2*x^2 - 8*I*a^2*c*d^2*x - 4*a^2*d^2 - (b^2*c^2*d^2*x^2 - 2*I*b^2*c*d^2*x - b^2*d^2)
*log(-(c*x + I)/(c*x - I))^2 + 4*(I*a*b*c^2*d^2*x^2 + 2*a*b*c*d^2*x - I*a*b*d^2)*log(-(c*x + I)/(c*x - I)))/x,
x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-12*b^2*c^4*d^2*integrate(1/16*x^4*arctan(c*x)^2/(c^2*x^3 + x), x) + 2*I*b^2*c^4*d^2*integrate(1/8*x^4*arctan(
c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - b^2*c^4*d^2*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x)
+ 2*I*b^2*c^4*d^2*integrate(1/8*x^4*arctan(c*x)/(c^2*x^3 + x), x) - 32*a*b*c^4*d^2*integrate(1/16*x^4*arctan(c
*x)/(c^2*x^3 + x), x) - 2*b^2*c^4*d^2*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 1/2*a^2*c^2*d^2*
x^2 + 12*I*b^2*c^3*d^2*integrate(1/8*x^3*arctan(c*x)^2/(c^2*x^3 + x), x) + 8*b^2*c^3*d^2*integrate(1/16*x^3*ar
ctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + I*b^2*c^3*d^2*integrate(1/8*x^3*log(c^2*x^2 + 1)^2/(c^2*x^3 + x
), x) + 20*b^2*c^3*d^2*integrate(1/16*x^3*arctan(c*x)/(c^2*x^3 + x), x) + 5*I*b^2*c^3*d^2*integrate(1/8*x^3*lo
g(c^2*x^2 + 1)/(c^2*x^3 + x), x) + 1/2*I*b^2*d^2*arctan(c*x)^3 - 8*I*b^2*c^2*d^2*integrate(1/8*x^2*arctan(c*x)
/(c^2*x^3 + x), x) + 2*I*a^2*c*d^2*x + 8*b^2*c*d^2*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x)
, x) + I*b^2*c*d^2*integrate(1/8*x*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 1/8*b^2*d^2*log(c^2*x^2 + 1)^2 + 2*I
*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^2 + 12*b^2*d^2*integrate(1/16*arctan(c*x)^2/(c^2*x^3 + x), x) -
2*I*b^2*d^2*integrate(1/8*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + b^2*d^2*integrate(1/16*log(c^2*x^2
+ 1)^2/(c^2*x^3 + x), x) + 32*a*b*d^2*integrate(1/16*arctan(c*x)/(c^2*x^3 + x), x) + a^2*d^2*log(x) - 1/8*(b^2
*c^2*d^2*x^2 - 4*I*b^2*c*d^2*x)*arctan(c*x)^2 + 1/8*(-I*b^2*c^2*d^2*x^2 - 4*b^2*c*d^2*x)*arctan(c*x)*log(c^2*x
^2 + 1) + 1/32*(b^2*c^2*d^2*x^2 - 4*I*b^2*c*d^2*x)*log(c^2*x^2 + 1)^2

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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