\(\int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx\) [19]

Optimal result

Integrand size = 18, antiderivative size = 499 \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\frac {i c (a+b \arctan (c x))^3}{c^2 d^2+e^2}+\frac {c^2 d (a+b \arctan (c x))^3}{e \left (c^2 d^2+e^2\right )}-\frac {(a+b \arctan (c x))^3}{e (d+e x)}-\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4974, 4968, 5104, 5004, 5040, 4964, 5114, 6745} \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^2} \, dx=\frac {3 i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}+\frac {3 i b^2 c \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d^2+e^2}-\frac {3 i b^2 c (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {i c (a+b \arctan (c x))^3}{c^2 d^2+e^2}+\frac {c^2 d (a+b \arctan (c x))^3}{e \left (c^2 d^2+e^2\right )}-\frac {3 b c \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2+e^2}+\frac {3 b c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2+e^2}+\frac {3 b c (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac {(a+b \arctan (c x))^3}{e (d+e x)}-\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )} \]

[In]

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

[Out]

(I*c*(a + b*ArcTan[c*x])^3)/(c^2*d^2 + e^2) + (c^2*d*(a + b*ArcTan[c*x])^3)/(e*(c^2*d^2 + e^2)) - (a + b*ArcTa
n[c*x])^3/(e*(d + e*x)) - (3*b*c*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (3*b*c*(a + b*Arc
Tan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2) + (3*b*c*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e
)*(1 - I*c*x))])/(c^2*d^2 + e^2) + ((3*I)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 +
e^2) + ((3*I)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2) - ((3*I)*b^2*c*(a + b*A
rcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) - (3*b^3*c*PolyLog[3, 1
- 2/(1 - I*c*x)])/(2*(c^2*d^2 + e^2)) + (3*b^3*c*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*(c^2*d^2 + e^2)) + (3*b^3*
c*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*(c^2*d^2 + e^2))

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 4968

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcTan[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + S
imp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] - Simp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1
- 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Si
mp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
NeQ[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 5104

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 22.85 (sec) , antiderivative size = 2398, normalized size of antiderivative = 4.81

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2398\)
default	\(\text {Expression too large to display}\)	\(2398\)
parts	\(\text {Expression too large to display}\)	\(2406\)

[In]

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

[Out]

1/c*(-a^3*c^2/(c*e*x+c*d)/e+b^3*c^2*(-1/(c*e*x+c*d)/e*arctan(c*x)^3+3/e*(arctan(c*x)^2*e/(c^2*d^2+e^2)*ln(c*e*
x+c*d)-1/2*arctan(c*x)^2/(c^2*d^2+e^2)*e*ln(c^2*x^2+1)+1/3*arctan(c*x)^3/(c^2*d^2+e^2)*d*c+e/(c^2*d^2+e^2)*arc
tan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-e/(c^2*d^2+e^2)*arctan(c*x)^2*ln(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(
1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)-I/(c^2*d^2+e^2)*e*c*d/(c*d-I*e)*arctan(c*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I
*c*x)^2/(c^2*x^2+1))+1/(c^2*d^2+e^2)*e*c*d/(c*d-I*e)*arctan(c*x)^2*ln(1-(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x
^2+1))+1/2/(c^2*d^2+e^2)*e*c*d/(c*d-I*e)*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-I*e^2*arctan(c
*x)*polylog(2,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)+e^2*arctan(c*x)^2*ln(1-(I*e
-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)+1/2*e^2*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*
x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)-1/3*I*e/(c^2*d^2+e^2)*arctan(c*x)^3+1/4*e/(c^2*d^2+e^2)*(I*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+(1+I*c*x)^2/(c^2*x^2+1))^2)^2+2*I*Pi*csgn(I*(-I*e*
(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^3+I*Pi*csgn(I*(1+(1+
I*c*x)^2/(c^2*x^2+1)))^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)+I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn
(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2-I*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*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2-I*
Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*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+(1+
I*c*x)^2/(c^2*x^2+1))^2)+I*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3-2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)
))*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2-I*
Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3+2*I*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*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3-2*I*Pi*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*
x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d))*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+
I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))^2+2*I*Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*(-I*e*(1+I*c*x)^2/(c
^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+c*d))*csgn(I*(-I*e*(1+I*c*x)^2/(c^2*x^2+1)+c*d*(1+I*c*x)^2/(c^2*x^2+
1)+I*e+c*d)/(1+(1+I*c*x)^2/(c^2*x^2+1)))+4*ln(2))*arctan(c*x)^2))+3*a*b^2*c^2*(-1/(c*e*x+c*d)/e*arctan(c*x)^2+
2/e*(arctan(c*x)*e/(c^2*d^2+e^2)*ln(c*e*x+c*d)-1/2*arctan(c*x)/(c^2*d^2+e^2)*e*ln(c^2*x^2+1)+1/2/(c^2*d^2+e^2)
*d*c*arctan(c*x)^2-e^2/(c^2*d^2+e^2)*(1/2*I*ln(c*e*x+c*d)*(-ln((I*e-c*e*x)/(c*d+I*e))+ln((I*e+c*e*x)/(I*e-c*d)
))/e-1/2*I*(dilog((I*e-c*e*x)/(c*d+I*e))-dilog((I*e+c*e*x)/(I*e-c*d)))/e)+1/2*e/(c^2*d^2+e^2)*(-1/2*I*(ln(c*x-
I)*ln(c^2*x^2+1)-dilog(-1/2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)+1/2*I*(ln(c*x+I)*ln(c^2*x
^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I)^2))))+3*a^2*b*c^2*(-1/(c*e*x+c*d)/e*arcta
n(c*x)+1/e*(e/(c^2*d^2+e^2)*ln(c*e*x+c*d)+1/(c^2*d^2+e^2)*(-1/2*e*ln(c^2*x^2+1)+d*c*arctan(c*x)))))

Fricas [F]

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

[In]

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

[Out]

integral((b^3*arctan(c*x)^3 + 3*a*b^2*arctan(c*x)^2 + 3*a^2*b*arctan(c*x) + a^3)/(e^2*x^2 + 2*d*e*x + d^2), x)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*atan(c*x))**3/(d + e*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

3/2*((2*c*d*arctan(c*x)/(c^2*d^2*e + e^3) - log(c^2*x^2 + 1)/(c^2*d^2 + e^2) + 2*log(e*x + d)/(c^2*d^2 + e^2))
*c - 2*arctan(c*x)/(e^2*x + d*e))*a^2*b - a^3/(e^2*x + d*e) - 1/32*(4*b^3*arctan(c*x)^3 - 3*b^3*arctan(c*x)*lo
g(c^2*x^2 + 1)^2 - 32*(e^2*x + d*e)*integrate(1/32*(28*(b^3*c^2*e*x^2 + b^3*e)*arctan(c*x)^3 + 12*(8*a*b^2*c^2
*e*x^2 + b^3*c*e*x + b^3*c*d + 8*a*b^2*e)*arctan(c*x)^2 - 12*(b^3*c^2*e*x^2 + b^3*c^2*d*x)*arctan(c*x)*log(c^2
*x^2 + 1) - 3*(b^3*c*e*x + b^3*c*d - (b^3*c^2*e*x^2 + b^3*e)*arctan(c*x))*log(c^2*x^2 + 1)^2)/(c^2*e^3*x^4 + 2
*c^2*d*e^2*x^3 + 2*d*e^2*x + d^2*e + (c^2*d^2*e + e^3)*x^2), x))/(e^2*x + d*e)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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