\(\int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) [106]

Optimal result

Integrand size = 23, antiderivative size = 216 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4996, 4974, 4972, 641, 46, 209, 5004, 4964, 5114, 6745} \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (-c x+i)}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (-c x+i)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}+\frac {i b^2}{2 c^2 d^2 (-c x+i)} \]

[In]

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

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

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 4972

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*
ArcTan[c*x])/(e*(q + 1))), x] - Dist[b*(c/(e*(q + 1))), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

Rule 4974

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a
+ b*ArcTan[c*x])^p/(e*(q + 1))), x] - Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p -
1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && N
eQ[q, -1]

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.39 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {\frac {12 i a^2}{-i+c x}-12 i a^2 \arctan (c x)-6 a^2 \log \left (1+c^2 x^2\right )-6 i a b \left (4 \arctan (c x)^2-\cos (2 \arctan (c x))+2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-2 i \arctan (c x) \left (\cos (2 \arctan (c x))-2 \log \left (1+e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))\right )+i \sin (2 \arctan (c x))\right )+b^2 \left (-8 i \arctan (c x)^3+3 \cos (2 \arctan (c x))+6 i \arctan (c x) \cos (2 \arctan (c x))-6 \arctan (c x)^2 \cos (2 \arctan (c x))+12 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-12 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-3 i \sin (2 \arctan (c x))+6 \arctan (c x) \sin (2 \arctan (c x))+6 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{12 c^2 d^2} \]

[In]

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

[Out]

(((12*I)*a^2)/(-I + c*x) - (12*I)*a^2*ArcTan[c*x] - 6*a^2*Log[1 + c^2*x^2] - (6*I)*a*b*(4*ArcTan[c*x]^2 - Cos[
2*ArcTan[c*x]] + 2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*(Cos[2*ArcTan[c*x]] - 2*Log[1 + E^((
2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]]) + I*Sin[2*ArcTan[c*x]]) + b^2*((-8*I)*ArcTan[c*x]^3 + 3*Cos[2*ArcTa
n[c*x]] + (6*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 6*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 12*ArcTan[c*x]^2*Log[1 +
E^((2*I)*ArcTan[c*x])] - (12*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*PolyLog[3, -E^((2*I)*ArcTa
n[c*x])] - (3*I)*Sin[2*ArcTan[c*x]] + 6*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]
]))/(12*c^2*d^2)

Maple [C] (warning: unable to verify)

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

Time = 6.03 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.97

method	result	size
derivativedivides	\(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\)	\(857\)
default	\(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\)	\(857\)
parts	\(-\frac {i a^{2}}{d^{2} c^{2} \left (-c x +i\right )}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{2}}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2} c^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c^{2}}+\frac {i a b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2} c^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2} c^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2} c^{2}}+\frac {a b}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2} c^{2}}-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2} c^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2} c^{2}}\)	\(902\)

[In]

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

[Out]

1/c^2*(-1/4*I*a*b/d^2*ln(c^2*x^2+1)-1/2*a^2/d^2*ln(c^2*x^2+1)-1/2*I*a*b/d^2*ln(c*x-I)^2+b^2/d^2*(I*arctan(c*x)
^2/(c*x-I)-arctan(c*x)^2*ln(c*x-I)+arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-2/3*I*arctan(c*x)^3+1/2*I*Pi*
csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2-
1/2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+
I*c*x)^2/(c^2*x^2+1)))*arctan(c*x)^2-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3*arct
an(c*x)^2-1/2*I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*a
rctan(c*x)^2-I*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*arctan(c*x)^2+1/2*arctan(c*x)^2+
I*Pi*arctan(c*x)^2-2*I*arctan(c*x)*(c*x+I)/(4*c*x-4*I)-1/4*(c*x+I)/(c*x-I)-I*arctan(c*x)*polylog(2,-(1+I*c*x)^
2/(c^2*x^2+1))+1/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1)))+I*a*b/d^2*dilog(-1/2*I*(c*x+I))-2*a*b/d^2*arctan(c*x)*
ln(c*x-I)+1/8*I*a*b/d^2*ln(c^4*x^4+10*c^2*x^2+9)-1/4*a*b/d^2*arctan(1/2*c*x)+1/4*a*b/d^2*arctan(1/6*c^3*x^3+7/
6*c*x)+1/2*a*b/d^2*arctan(1/2*c*x-1/2*I)+a*b/d^2/(c*x-I)+I*a*b/d^2*ln(-1/2*I*(c*x+I))*ln(c*x-I)+1/2*a*b/d^2*ar
ctan(c*x)+I*a^2/d^2/(c*x-I)-I*a^2/d^2*arctan(c*x)+2*I*a*b/d^2*arctan(c*x)/(c*x-I))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

a^2*(I/(c^3*d^2*x - I*c^2*d^2) - log(c*x - I)/(c^2*d^2)) - 1/32*(-8*I*b^2*arctan(c*x)^2 - 8*(-I*b^2*c*x - b^2)
*arctan(c*x)^3 - (b^2*c*x - I*b^2)*log(c^2*x^2 + 1)^3 - 2*(-I*b^2 + (-I*b^2*c*x - b^2)*arctan(c*x))*log(c^2*x^
2 + 1)^2 - (2*b^2*c^3*(c^2/(c^9*d^2*x^2 + c^7*d^2) + log(c^2*x^2 + 1)/(c^7*d^2*x^2 + c^5*d^2)) - 640*b^2*c^3*i
ntegrate(1/16*x^3*arctan(c*x)^2/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 96*b^2*c^3*integrate(1/16*x^3*log(
c^2*x^2 + 1)^2/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 1024*a*b*c^3*integrate(1/16*x^3*arctan(c*x)/(c^5*d^
2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) - 256*b^2*c^2*integrate(1/16*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^5*d^2*x^4
+ 2*c^3*d^2*x^2 + c*d^2), x) + 256*b^2*c^2*integrate(1/16*x^2*arctan(c*x)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2
), x) + 16*(c*(x/(c^5*d^2*x^2 + c^3*d^2) + arctan(c*x)/(c^4*d^2)) - 2*arctan(c*x)/(c^5*d^2*x^2 + c^3*d^2))*a*b
*c + 128*b^2*c*integrate(1/16*x*arctan(c*x)^2/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) + b^2*c*log(c^2*x^2 +
1)^2/(c^5*d^2*x^2 + c^3*d^2) + 256*b^2*integrate(1/16*arctan(c*x)/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x))*(
c^3*d^2*x - I*c^2*d^2) - 32*(I*c^3*d^2*x + c^2*d^2)*integrate(-1/8*(32*a*b*c^2*x^2*arctan(c*x) - b^2*log(c^2*x
^2 + 1)^2 + 4*(2*b^2*c^2*x^2 - b^2)*arctan(c*x)^2 - 2*(b^2*c^2*x^2 + b^2 + (b^2*c^3*x^3 - b^2*c*x)*arctan(c*x)
)*log(c^2*x^2 + 1))/(c^5*d^2*x^4 + 2*c^3*d^2*x^2 + c*d^2), x) + 4*(2*b^2*arctan(c*x) - (b^2*c*x - I*b^2)*arcta
n(c*x)^2)*log(c^2*x^2 + 1))/(c^3*d^2*x - I*c^2*d^2)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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