\(\int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx\) [127]

Optimal result

Integrand size = 25, antiderivative size = 410 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d} \]

[Out]

-3/2*b*(a+b*arctan(c*x))^2/c^3/d+3/2*I*b*x*(a+b*arctan(c*x))^2/c^2/d+1/2*I*(a+b*arctan(c*x))^3/c^3/d+x*(a+b*ar
ctan(c*x))^3/c^2/d-1/2*I*x^2*(a+b*arctan(c*x))^3/c/d+3*I*b^2*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3/d+3*b*(a+b*
arctan(c*x))^2*ln(2/(1+I*c*x))/c^3/d-I*(a+b*arctan(c*x))^3*ln(2/(1+I*c*x))/c^3/d-3/2*b^3*polylog(2,1-2/(1+I*c*
x))/c^3/d+3*I*b^2*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^3/d+3/2*b*(a+b*arctan(c*x))^2*polylog(2,1-2/(1+
I*c*x))/c^3/d+3/2*b^3*polylog(3,1-2/(1+I*c*x))/c^3/d-3/2*I*b^2*(a+b*arctan(c*x))*polylog(3,1-2/(1+I*c*x))/c^3/
d-3/4*b^3*polylog(4,1-2/(1+I*c*x))/c^3/d

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4986, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 5004, 5114, 6745, 5118} \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^3 d}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c^3 d}+\frac {3 i b^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c^3 d}-\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {3 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^3 d}-\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c^3 d} \]

[In]

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

[Out]

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

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 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 4986

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[f/e,
Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f/e), 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] && GtQ[m, 0]

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 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 5118

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

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {i \left (4 i a^3 c x-6 a^2 b c x+2 a^3 c^2 x^2-4 i a^3 \arctan (c x)+6 a^2 b \arctan (c x)+12 i a^2 b c x \arctan (c x)-12 a b^2 c x \arctan (c x)+6 a^2 b c^2 x^2 \arctan (c x)-12 i a^2 b \arctan (c x)^2+18 a b^2 \arctan (c x)^2+6 i b^3 \arctan (c x)^2+12 i a b^2 c x \arctan (c x)^2-6 b^3 c x \arctan (c x)^2+6 a b^2 c^2 x^2 \arctan (c x)^2-8 i a b^2 \arctan (c x)^3+6 b^3 \arctan (c x)^3+4 i b^3 c x \arctan (c x)^3+2 b^3 c^2 x^2 \arctan (c x)^3-2 i b^3 \arctan (c x)^4+12 a^2 b \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-12 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+12 a b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+12 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+4 b^3 \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )-2 a^3 \log \left (1+c^2 x^2\right )-6 i a^2 b \log \left (1+c^2 x^2\right )+6 a b^2 \log \left (1+c^2 x^2\right )-6 i b (a+i b+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 b^2 (a+i b+b \arctan (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right )}{4 c^3 d} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 14.23 (sec) , antiderivative size = 1359, normalized size of antiderivative = 3.31

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1359\)
default	\(\text {Expression too large to display}\)	\(1359\)
parts	\(\text {Expression too large to display}\)	\(1406\)

[In]

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

[Out]

1/c^3*(a^3/d*c*x-3/2*I*a^2*b/d*arctan(c*x)*c^2*x^2+3/2*I*a^2*b/d*c*x-a^3/d*arctan(c*x)+b^3/d*(-1/2*I*arctan(c*
x)^2*(I*arctan(c*x)+arctan(c*x)*c*x-3)*(I+c*x)-1/2*arctan(c*x)^4-I*arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)+1)
-3/2*arctan(c*x)^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*arctan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+3
/4*polylog(4,-(1+I*c*x)^2/(c^2*x^2+1))+3*arctan(c*x)^2+3*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+3/2*polyl
og(2,-(1+I*c*x)^2/(c^2*x^2+1))-2*I*arctan(c*x)^3+3*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-3*I*arctan(c*x)
*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1)))+3*a*b^2/d*(arctan(c*x)^2*c*x-1/2
*I*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-3/2*I*arctan(c*x)^2+I*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-Pi*csgn((1+I*c*x)^2
/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I
*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+Pi*arctan(c*x)^2+2*arctan(c*x)*ln(1+I*(1+I*c*
x)/(c^2*x^2+1)^(1/2))+2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*(c*x-I)-1/2*Pi*csgn((1+I
*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-1/2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*cs
gn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+I*arctan(c
*x)^2*ln(c*x-I)-2*I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*arctan(c*x)^2*c^2*x^2-1/2*Pi*csgn((1+I*c*x)^2
/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-I*arctan(c*x)^2*ln(2*I
*(1+I*c*x)^2/(c^2*x^2+1))-2/3*arctan(c*x)^3-arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2*I*dilog(1+I*(1+I
*c*x)/(c^2*x^2+1)^(1/2)))+3*a^2*b/d*arctan(c*x)*c*x+1/2*I*a^3/d*ln(c^2*x^2+1)+3/4*I*a^2*b/d*arctan(1/2*c*x-1/2
*I)-3/4*a^2*b/d*ln(c*x-I)^2+3/2*a^2*b/d*ln(c*x-I)*ln(-1/2*I*(I+c*x))+3/2*a^2*b/d*dilog(-1/2*I*(I+c*x))+3*I*a^2
*b/d*arctan(c*x)*ln(c*x-I)+3/2*a^2*b/d-9/8*a^2*b/d*ln(c^2*x^2+1)+3/8*I*a^2*b/d*arctan(1/6*c^3*x^3+7/6*c*x)-3/1
6*a^2*b/d*ln(c^4*x^4+10*c^2*x^2+9)-9/4*I*a^2*b/d*arctan(c*x)-1/2*I*a^3/d*c^2*x^2-3/8*I*a^2*b/d*arctan(1/2*c*x)
)

Fricas [F]

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

[In]

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

[Out]

integral(-1/8*(b^3*x^2*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*x^2*log(-(c*x + I)/(c*x - I))^2 - 12*a^2*b*x^2*
log(-(c*x + I)/(c*x - I)) + 8*I*a^3*x^2)/(c*d*x - I*d), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*a^3*((I*c*x^2 - 2*x)/(c^2*d) - 2*I*log(I*c*x + 1)/(c^3*d)) - 1/128*(16*b^3*arctan(c*x)^4 - b^3*log(c^2*x^
2 + 1)^4 + 4*(384*b^3*c^3*integrate(1/64*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) - 32*b^3*c
^3*integrate(1/64*x^3*log(c^2*x^2 + 1)^3/(c^4*d*x^2 + c^2*d), x) + 384*b^3*c^3*integrate(1/64*x^3*arctan(c*x)^
2/(c^4*d*x^2 + c^2*d), x) - 96*b^3*c^3*integrate(1/64*x^3*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 1792*b^
3*c^2*integrate(1/64*x^2*arctan(c*x)^3/(c^4*d*x^2 + c^2*d), x) - 192*b^3*c^2*integrate(1/64*x^2*arctan(c*x)*lo
g(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 6144*a*b^2*c^2*integrate(1/64*x^2*arctan(c*x)^2/(c^4*d*x^2 + c^2*d)
, x) - 384*b^3*c^2*integrate(1/64*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) - 6144*a^2*b*c^2*in
tegrate(1/64*x^2*arctan(c*x)/(c^4*d*x^2 + c^2*d), x) + 384*b^3*c*integrate(1/64*x*arctan(c*x)^2*log(c^2*x^2 +
1)/(c^4*d*x^2 + c^2*d), x) + 96*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^3/(c^4*d*x^2 + c^2*d), x) + 768*b^3*c*
integrate(1/64*x*arctan(c*x)^2/(c^4*d*x^2 + c^2*d), x) - 192*b^3*c*integrate(1/64*x*log(c^2*x^2 + 1)^2/(c^4*d*
x^2 + c^2*d), x) - 192*b^3*integrate(1/64*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^4*d*x^2 + c^2*d), x) - 3*b^3*arcta
n(c*x)^4/(c^3*d))*c^3*d - 128*I*c^3*d*integrate(-1/64*(192*a^2*b*c^3*x^3*arctan(c*x) + 8*(7*b^3*c^3*x^3 - 3*b^
3*c*x)*arctan(c*x)^3 - (b^3*c^2*x^2 + 3*b^3)*log(c^2*x^2 + 1)^3 + 12*(16*a*b^2*c^3*x^3 + b^3*c^2*x^2)*arctan(c
*x)^2 - 3*(b^3*c^2*x^2 - 2*(b^3*c^3*x^3 - b^3*c*x)*arctan(c*x))*log(c^2*x^2 + 1)^2 + 12*((b^3*c^2*x^2 - b^3)*a
rctan(c*x)^2 + (b^3*c^3*x^3 + 2*b^3*c*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^4*d*x^2 + c^2*d), x) + 8*(I*b^3*c^2
*x^2 - 2*b^3*c*x)*arctan(c*x)^3 + 6*(-I*b^3*c^2*x^2 + 2*b^3*c*x)*arctan(c*x)*log(c^2*x^2 + 1)^2 + (b^3*c^2*x^2
+ 2*I*b^3*c*x + 4*I*b^3*arctan(c*x))*log(c^2*x^2 + 1)^3 + 4*(4*I*b^3*arctan(c*x)^3 - 3*(b^3*c^2*x^2 + 2*I*b^3
*c*x)*arctan(c*x)^2)*log(c^2*x^2 + 1))/(c^3*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]

[In]

int((x^2*(a + b*atan(c*x))^3)/(d + c*d*x*1i),x)

[Out]

int((x^2*(a + b*atan(c*x))^3)/(d + c*d*x*1i), x)