\(\int (d+i c d x) (a+b \arctan (c x))^3 \, dx\) [122]

Optimal result

Integrand size = 20, antiderivative size = 220 \[ \int (d+i c d x) (a+b \arctan (c x))^3 \, dx=\frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4974, 4930, 5040, 4964, 2449, 2352, 1600, 5004, 5112, 6745} \[ \int (d+i c d x) (a+b \arctan (c x))^3 \, dx=-\frac {3 i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c}-\frac {3 i b^2 d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 c} \]

[In]

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

[Out]

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

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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 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 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 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 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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[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}& = -\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {(3 i b) \int \left (-d^2 (a+b \arctan (c x))^2-\frac {2 i \left (i d^2-c d^2 x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2}\right ) \, dx}{2 d} \\ & = -\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {(3 b) \int \frac {\left (i d^2-c d^2 x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{d}-\frac {1}{2} (3 i b d) \int (a+b \arctan (c x))^2 \, dx \\ & = -\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {(3 b) \int \frac {(a+b \arctan (c x))^2}{-\frac {i}{d^2}-\frac {c x}{d^2}} \, dx}{d}+\left (3 i b^2 c d\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\left (3 i b^2 d\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx-\left (6 b^2 d\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx \\ & = \frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\left (3 i b^3 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 i b^3 d\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx \\ & = \frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 c}+\frac {\left (3 b^3 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c} \\ & = \frac {3 b d (a+b \arctan (c x))^2}{2 c}-\frac {3}{2} i b d x (a+b \arctan (c x))^2-\frac {i d (1+i c x)^2 (a+b \arctan (c x))^3}{2 c}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^2 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.67 \[ \int (d+i c d x) (a+b \arctan (c x))^3 \, dx=\frac {i d \left (-2 i a^3 c x-3 a^2 b c x+a^3 c^2 x^2+3 a^2 b \arctan (c x)-6 i a^2 b c x \arctan (c x)-6 a b^2 c x \arctan (c x)+3 a^2 b c^2 x^2 \arctan (c x)-3 a b^2 \arctan (c x)^2+3 i b^3 \arctan (c x)^2-6 i a b^2 c x \arctan (c x)^2-3 b^3 c x \arctan (c x)^2+3 a b^2 c^2 x^2 \arctan (c x)^2-b^3 \arctan (c x)^3-2 i b^3 c x \arctan (c x)^3+b^3 c^2 x^2 \arctan (c x)^3-12 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-6 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-6 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+3 i a^2 b \log \left (1+c^2 x^2\right )+3 a b^2 \log \left (1+c^2 x^2\right )-3 b^2 (2 a-i b+2 b \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 i b^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{2 c} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 2.04 (sec) , antiderivative size = 3777, normalized size of antiderivative = 17.17

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3777\)
default	\(\text {Expression too large to display}\)	\(3777\)
parts	\(\text {Expression too large to display}\)	\(3779\)

[In]

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

[Out]

1/c*(-I*d*a^3*(-1/2*c^2*x^2+I*c*x)+d*b^3*(1/2*I*arctan(c*x)^3*c^2*x^2+arctan(c*x)^3*c*x+3/2*I*(-1/4*Pi*csgn(I*
(1+I*c*x)^2/(c^2*x^2+1))^3*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)
^2/(c^2*x^2+1)))-1/4*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*(2*I*arctan(c*x)*ln((1
+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+1/4*Pi*csgn(I*((1+I*c*x)^2/(c^2*
x^2+1)+1)^2)^3*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+
1)))-1/2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arcta
n(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2
+1)^(1/2)))+1/2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arct
an(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^
2+1)^(1/2)))-2*ln(2)*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*ln(2)*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2
*x^2+1)^(1/2))+2*ln(2)*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*I*ln(2)*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/
2))+2*I*ln(2)*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*arctan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-I*ln(2)
*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*(I
*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(
1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+I*ln(c^2*x^2+1)*arctan(c*x)^2-2*I*ln(2)*ar
ctan(c*x)^2-1/3*arctan(c*x)^3-arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-arctan(c*x)*ln(1-I*(1+I*c*x)/(c^
2*x^2+1)^(1/2))-2*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-arctan(c*x)^2*c*x-1/4*Pi*csgn(I*(1+I*c*x)^2/
(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^
2)*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-arctan(
c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-1/2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1
)+1)^2)*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+d
ilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+1/2*Pi*csgn(I*(1+I*c*x)/(c^2*x^2
+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+po
lylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-1/2*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)^2)^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-1/
4*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^
2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+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)^2)^2*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I
*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+1/4*Pi
*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*(2*I*ar
ctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+1/4*Pi*csgn(I*((1
+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+
1)+2*arctan(c*x)^2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))+1/4*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x
)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*(2*I*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*arctan(c*x)^
2+polylog(2,-(1+I*c*x)^2/(c^2*x^2+1)))-1/2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(
(1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c
*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))+1/2*Pi*c
sgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+
1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(
1+I*c*x)/(c^2*x^2+1)^(1/2)))-Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*(I*arcta
n(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*
x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))-1/2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)
^2/(c^2*x^2+1)+1)^2)^2*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2)
)+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)
/(c^2*x^2+1)^(1/2)))+I*arctan(c*x)^2+I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1
)^(1/2))-I*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*I*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*Pi*csgn(I*(1+I*c*
x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I/((1+I*c*x)^2/(c^2*x^2+1
)+1)^2)*(I*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+d
ilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2)))))+3*d*a*b^2*(1/2*I*arctan(c*x)^2
*c^2*x^2+arctan(c*x)^2*c*x+I*(I*ln(c^2*x^2+1)*arctan(c*x)+1/2*arctan(c*x)^2-arctan(c*x)*c*x-1/2*ln(c*x-I)*ln(c
^2*x^2+1)+1/4*ln(c*x-I)^2+1/2*dilog(-1/2*I*(I+c*x))+1/2*ln(c*x-I)*ln(-1/2*I*(I+c*x))+1/2*ln(I+c*x)*ln(c^2*x^2+
1)-1/4*ln(I+c*x)^2-1/2*dilog(1/2*I*(c*x-I))-1/2*ln(I+c*x)*ln(1/2*I*(c*x-I))+1/2*ln(c^2*x^2+1)))+3*d*a^2*b*(1/2
*I*arctan(c*x)*c^2*x^2+arctan(c*x)*c*x+1/2*I*(-c*x+I*ln(c^2*x^2+1)+arctan(c*x))))

Fricas [F]

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

[In]

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

[Out]

1/16*(b^3*c*d*x^2 - 2*I*b^3*d*x)*log(-(c*x + I)/(c*x - I))^3 + integral(1/8*(8*I*a^3*c^3*d*x^3 + 8*a^3*c^2*d*x
^2 + 8*I*a^3*c*d*x + 8*a^3*d - 3*(2*I*a*b^2*c^3*d*x^3 + (2*a*b^2 - I*b^3)*c^2*d*x^2 + 2*a*b^2*d + 2*(I*a*b^2 -
b^3)*c*d*x)*log(-(c*x + I)/(c*x - I))^2 - 12*(a^2*b*c^3*d*x^3 - I*a^2*b*c^2*d*x^2 + a^2*b*c*d*x - I*a^2*b*d)*
log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

Sympy [F(-2)]

Exception generated. \[ \int (d+i c d x) (a+b \arctan (c x))^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

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

[In]

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

[Out]

12*b^3*c^3*d*integrate(1/64*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - b^3*c^3*d*integrate(1/64*x^
3*log(c^2*x^2 + 1)^3/(c^2*x^2 + 1), x) + 12*b^3*c^3*d*integrate(1/64*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) - 3*b
^3*c^3*d*integrate(1/64*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1/2*I*a^3*c*d*x^2 + 7/32*b^3*d*arctan(c*x)^
4/c + 56*b^3*c^2*d*integrate(1/64*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 6*b^3*c^2*d*integrate(1/64*x^2*arctan(
c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2*d*integrate(1/64*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x)
+ 36*b^3*c^2*d*integrate(1/64*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3/2*I*(x^2*arctan(c*x) - c
*(x/c^2 - arctan(c*x)/c^3))*a^2*b*c*d + a*b^2*d*arctan(c*x)^3/c + 12*b^3*c*d*integrate(1/64*x*arctan(c*x)^2*lo
g(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - b^3*c*d*integrate(1/64*x*log(c^2*x^2 + 1)^3/(c^2*x^2 + 1), x) - 24*b^3*c*d*
integrate(1/64*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 6*b^3*c*d*integrate(1/64*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1)
, x) + a^3*d*x + 6*b^3*d*integrate(1/64*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(2*c*x*arctan(c
*x) - log(c^2*x^2 + 1))*a^2*b*d/c + 1/16*(I*b^3*c*d*x^2 + 2*b^3*d*x)*arctan(c*x)^3 - 3/32*(b^3*c*d*x^2 - 2*I*b
^3*d*x)*arctan(c*x)^2*log(c^2*x^2 + 1) + 3/64*(-I*b^3*c*d*x^2 - 2*b^3*d*x)*arctan(c*x)*log(c^2*x^2 + 1)^2 + 1/
128*(b^3*c*d*x^2 - 2*I*b^3*d*x)*log(c^2*x^2 + 1)^3 + I*integrate(1/64*(56*(b^3*c^3*d*x^3 + b^3*c*d*x)*arctan(c
*x)^3 + (b^3*c^2*d*x^2 + b^3*d)*log(c^2*x^2 + 1)^3 + 12*(16*a*b^2*c^3*d*x^3 - 3*b^3*c^2*d*x^2 + 16*a*b^2*c*d*x
)*arctan(c*x)^2 + 3*(3*b^3*c^2*d*x^2 + 2*(b^3*c^3*d*x^3 + b^3*c*d*x)*arctan(c*x))*log(c^2*x^2 + 1)^2 - 12*((b^
3*c^2*d*x^2 + b^3*d)*arctan(c*x)^2 - (b^3*c^3*d*x^3 - 2*b^3*c*d*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1
), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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