3.19 \(\int \frac {(a+b \tan ^{-1}(c x))^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=499 \[ \frac {3 i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {3 b c \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac {3 b c \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )} \]

[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]  time = 0.53, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4864, 4858, 4984, 4884, 4920, 4854, 4994, 6610} \[ \frac {3 i b^2 c \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {3 b c \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac {3 b c \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)} \]

Antiderivative was successfully verified.

[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 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 4858

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] + Sim
p[(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] + Simp
[(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] && Ne
Q[c^2*d^2 + e^2, 0]

Rule 4864

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 4884

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 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4984

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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 6610

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*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{(d+e x)^2} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {(3 b c) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {c^2 (d-e x) \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}+\frac {\left (3 b c^3\right ) \int \frac {(d-e x) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )}+\frac {(3 b c e) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{c^2 d^2+e^2}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}+\frac {\left (3 b c^3 d\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )}\\ &=\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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 {\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2 d^2+e^2}\\ &=\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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 {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}\\ &=\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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 {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}\\ &=\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^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 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (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 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 b^3 c \text {Li}_3\left (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*}

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Mathematica [A]  time = 21.65, size = 770, normalized size = 1.54 \[ -\frac {a^3}{e (d+e x)}+\frac {3 a^2 b c^2 d \tan ^{-1}(c x)}{c^2 d^2 e+e^3}-\frac {3 a^2 b c \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {3 a^2 b c \log (d+e x)}{c^2 d^2+e^2}-\frac {3 a^2 b \tan ^{-1}(c x)}{e (d+e x)}+\frac {3 a b^2 \left (-\frac {c d \left (-\frac {1}{2} \pi \log \left (c^2 x^2+1\right )+i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-i \tan ^{-1}(c x) \left (\pi -2 \tan ^{-1}\left (\frac {c d}{e}\right )\right )-2 \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\pi \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )\right )}{c^2 d^2+e^2}-\frac {\tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}}{e \sqrt {\frac {c^2 d^2}{e^2}+1}}+\frac {x \tan ^{-1}(c x)^2}{d+e x}\right )}{d}+\frac {b^3 \left (\frac {2 x \tan ^{-1}(c x)^3}{d+e x}+\frac {2 \tan ^{-1}(c x) \left (\tan ^{-1}(c x)^2 \left (-2 e \sqrt {\frac {c^2 d^2}{e^2}+1} e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}+i c d+e\right )+3 c d \left (2 \tan ^{-1}\left (\frac {c d}{e}\right ) \left (\log \left (\frac {e^{-i \tan ^{-1}\left (\frac {c d}{e}\right )} \left ((c x-i) e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )}+c x+i\right )}{2 \sqrt {c^2 x^2+1}}\right )+\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )+\pi \left (\log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-\log \left (-\frac {2 i}{c x-i}\right )\right )\right )+3 c d \tan ^{-1}(c x) \left (2 \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-\log \left (-i e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \sin \left (2 \tan ^{-1}(c x)\right )-e^{2 i \tan ^{-1}\left (\frac {c d}{e}\right )} \cos \left (2 \tan ^{-1}(c x)\right )+1\right )\right )\right )-6 i c d \tan ^{-1}(c x) \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+3 c d \text {Li}_3\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )}{c^2 d^2+e^2}\right )}{2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^3/(e*(d + e*x))) + (3*a^2*b*c^2*d*ArcTan[c*x])/(c^2*d^2*e + e^3) - (3*a^2*b*ArcTan[c*x])/(e*(d + e*x)) + (
3*a^2*b*c*Log[d + e*x])/(c^2*d^2 + e^2) - (3*a^2*b*c*Log[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)) + (3*a*b^2*(-((E^(I
*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e)) + (x*ArcTan[c*x]^2)/(d + e*x) - (c*d*((-I)*(Pi -
2*ArcTan[(c*d)/e])*ArcTan[c*x] - Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*(ArcTan[(c*d)/e] + ArcTan[c*x])*Log[1
- E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - (Pi*Log[1 + c^2*x^2])/2 + 2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*
d)/e] + ArcTan[c*x]]] + I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/d + (b^3*(
(2*x*ArcTan[c*x]^3)/(d + e*x) + (2*ArcTan[c*x]*((I*c*d + e - 2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])
)*ArcTan[c*x]^2 + 3*c*d*ArcTan[c*x]*(2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - Log[1 - E^((2*I)*A
rcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]]) + 3*c*d*(Pi*(Log[1 + E^(
(-2*I)*ArcTan[c*x])] - Log[(-2*I)/(-I + c*x)]) + 2*ArcTan[(c*d)/e]*(Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan
[c*x]))] + Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] -
Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - Log[
Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]))) - (6*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c
*x]))] + 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(c^2*d^2 + e^2)))/(2*d)

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arctan \left (c x\right )^{3} + 3 \, a b^{2} \arctan \left (c x\right )^{2} + 3 \, a^{2} b \arctan \left (c x\right ) + a^{3}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [C]  time = 1.23, size = 2960, normalized size = 5.93 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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))*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*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*Pi*c
sgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*
csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))+3*c^2*b
^3/(c^2*d^2+e^2)*d/(d*c-I*e)*arctan(c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+3*c*b^3*e*arctan(
c*x)^2*ln(1-(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)+3*c^2*a^2*b/e/(c^2*d^2+e^2)*d
*arctan(c*x)+3*c^2*a*b^2/e/(c^2*d^2+e^2)*d*arctan(c*x)^2+3/2*I*c*a*b^2/(c^2*d^2+e^2)*ln(c*x-I)*ln(-1/2*I*(I+c*
x))+3/2*I*c*a*b^2/(c^2*d^2+e^2)*ln(I+c*x)*ln(c^2*x^2+1)-3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*Pi*csgn(I*(1+I
*c*x)^2/(c^2*x^2+1))^3+3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3+3/2*
I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*
c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3+3*I*c*a*b^2/(c^2*d^2+e^2)*ln(c*e*x+c*d)*ln((I*e-c*e*x)/(d*c+I*e))-3/2*I*c*a*
b^2/(c^2*d^2+e^2)*ln(I+c*x)*ln(1/2*I*(c*x-I))-3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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)^3-3*I*c*a*b^2/(c^2*d^2+e^2)*ln(c*e*x+c*d)*ln((I*e+c*e*x)/(I*e-d*c))-3
/2*I*c*a*b^2/(c^2*d^2+e^2)*ln(c*x-I)*ln(c^2*x^2+1)+3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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)+3/2*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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+3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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)^2-3*I*c^2*b^3/(c^2*d
^2+e^2)*d/(d*c-I*e)*arctan(c*x)*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))-3*I*c*b^3*e*arctan(c*x)
*polylog(2,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)-3/2*I*c*b^3/(c^2*d^2+e^2)*arct
an(c*x)^2*Pi*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c))*csgn(I*(-I*(1+I*c*x)^2
/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-3/2*I*c*b^3/(c^2*d^2+e^2)*a
rctan(c*x)^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*
x^2+1)+I*e+d*c)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-3/2*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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)^2-3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*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))+3/4*I*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^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+I*c*x)^2/(c^2*x^2+1)+1)^2)^2-c*a^3/(c*e*x+c*d
)/e+3*c*a^2*b/(c^2*d^2+e^2)*ln(c*e*x+c*d)-3/2*c*a^2*b/(c^2*d^2+e^2)*ln(c^2*x^2+1)+3*c*b^3/(c^2*d^2+e^2)*arctan
(c*x)^2*ln(2)+3*c*b^3*arctan(c*x)^2/(c^2*d^2+e^2)*ln(c*e*x+c*d)-3/2*c*b^3*arctan(c*x)^2/(c^2*d^2+e^2)*ln(c^2*x
^2+1)-c*b^3/(c*e*x+c*d)/e*arctan(c*x)^3+3*c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-3*
c*b^3/(c^2*d^2+e^2)*arctan(c*x)^2*ln(-I*(1+I*c*x)^2/(c^2*x^2+1)*e+c*d*(1+I*c*x)^2/(c^2*x^2+1)+I*e+d*c)-I*c*b^3
/(c^2*d^2+e^2)*arctan(c*x)^3+c^2*b^3/e*arctan(c*x)^3/(c^2*d^2+e^2)*d-3*c*a*b^2/(c*e*x+c*d)/e*arctan(c*x)^2+6*c
*a*b^2*arctan(c*x)/(c^2*d^2+e^2)*ln(c*e*x+c*d)-3*c*a*b^2*arctan(c*x)/(c^2*d^2+e^2)*ln(c^2*x^2+1)-3*c*a^2*b/(c*
e*x+c*d)/e*arctan(c*x)+3/2*c^2*b^3/(c^2*d^2+e^2)*d/(d*c-I*e)*polylog(3,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^
2+1))+3/2*c*b^3*e*polylog(3,(I*e-d*c)/(d*c+I*e)*(1+I*c*x)^2/(c^2*x^2+1))/(c^2*d^2+e^2)/(e+I*d*c)-3*I*c*a*b^2/(
c^2*d^2+e^2)*dilog((I*e+c*e*x)/(I*e-d*c))-3/2*I*c*a*b^2/(c^2*d^2+e^2)*dilog(1/2*I*(c*x-I))+3*I*c*a*b^2/(c^2*d^
2+e^2)*dilog((I*e-c*e*x)/(d*c+I*e))-3/4*I*c*a*b^2/(c^2*d^2+e^2)*ln(I+c*x)^2+3/4*I*c*a*b^2/(c^2*d^2+e^2)*ln(c*x
-I)^2+3/2*I*c*a*b^2/(c^2*d^2+e^2)*dilog(-1/2*I*(I+c*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{2} \, {\left ({\left (\frac {2 \, c d \arctan \left (c x\right )}{c^{2} d^{2} e + e^{3}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{2} d^{2} + e^{2}} + \frac {2 \, \log \left (e x + d\right )}{c^{2} d^{2} + e^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{e^{2} x + d e}\right )} a^{2} b - \frac {a^{3}}{e^{2} x + d e} - \frac {\frac {15}{2} \, b^{3} \arctan \left (c x\right )^{3} - \frac {21}{8} \, b^{3} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2} - {\left (e^{2} x + d e\right )} \int \frac {196 \, {\left (b^{3} c^{2} e x^{2} + b^{3} e\right )} \arctan \left (c x\right )^{3} + 12 \, {\left (64 \, a b^{2} c^{2} e x^{2} + 15 \, b^{3} c e x + 15 \, b^{3} c d + 64 \, a b^{2} e\right )} \arctan \left (c x\right )^{2} - 84 \, {\left (b^{3} c^{2} e x^{2} + b^{3} c^{2} d x\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) - 21 \, {\left (b^{3} c e x + b^{3} c d - {\left (b^{3} c^{2} e x^{2} + b^{3} e\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{8 \, {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e + {\left (c^{2} d^{2} e + e^{3}\right )} x^{2}\right )}}\,{d x}}{32 \, {\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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