\(\int \frac {x^3 (a+b \arctan (c x))^2}{d+e x} \, dx\) [141]

Optimal result

Integrand size = 21, antiderivative size = 598 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+e x} \, dx=\frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \arctan (c x)}{3 c^3 e}+\frac {b^2 d x \arctan (c x)}{c e^2}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2 e^2}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c e^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3 e}+\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {4996, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 327, 209, 4968} \[ \int \frac {x^3 (a+b \arctan (c x))^2}{d+e x} \, dx=-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}-\frac {2 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3 e}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^4}+\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}+\frac {d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{e^4}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^4}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}+\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {a b d x}{c e^2}-\frac {b^2 \arctan (c x)}{3 c^3 e}+\frac {b^2 d x \arctan (c x)}{c e^2}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3 e}-\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2 e^2}+\frac {b^2 x}{3 c^2 e}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c e^3} \]

[In]

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

[Out]

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

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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 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 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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 (a+b \arctan (c x))^2}{e^3}-\frac {d x (a+b \arctan (c x))^2}{e^2}+\frac {x^2 (a+b \arctan (c x))^2}{e}-\frac {d^3 (a+b \arctan (c x))^2}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d^2 \int (a+b \arctan (c x))^2 \, dx}{e^3}-\frac {d^3 \int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx}{e^3}-\frac {d \int x (a+b \arctan (c x))^2 \, dx}{e^2}+\frac {\int x^2 (a+b \arctan (c x))^2 \, dx}{e} \\ & = \frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 b c d^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{e^3}+\frac {(b c d) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{e^2}-\frac {(2 b c) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 e} \\ & = \frac {i d^2 (a+b \arctan (c x))^2}{c e^3}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {\left (2 b d^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{e^3}+\frac {(b d) \int (a+b \arctan (c x)) \, dx}{c e^2}-\frac {(b d) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c e^2}-\frac {(2 b) \int x (a+b \arctan (c x)) \, dx}{3 c e}+\frac {(2 b) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c e} \\ & = \frac {a b d x}{c e^2}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{e^3}+\frac {\left (b^2 d\right ) \int \arctan (c x) \, dx}{c e^2}+\frac {b^2 \int \frac {x^2}{1+c^2 x^2} \, dx}{3 e}-\frac {(2 b) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2 e} \\ & = \frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}+\frac {b^2 d x \arctan (c x)}{c e^2}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}+\frac {\left (2 i b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c e^3}-\frac {\left (b^2 d\right ) \int \frac {x}{1+c^2 x^2} \, dx}{e^2}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2 e}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2 e} \\ & = \frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \arctan (c x)}{3 c^3 e}+\frac {b^2 d x \arctan (c x)}{c e^2}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2 e^2}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c e^3}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3 e} \\ & = \frac {a b d x}{c e^2}+\frac {b^2 x}{3 c^2 e}-\frac {b^2 \arctan (c x)}{3 c^3 e}+\frac {b^2 d x \arctan (c x)}{c e^2}-\frac {b x^2 (a+b \arctan (c x))}{3 c e}+\frac {i d^2 (a+b \arctan (c x))^2}{c e^3}-\frac {d (a+b \arctan (c x))^2}{2 c^2 e^2}-\frac {i (a+b \arctan (c x))^2}{3 c^3 e}+\frac {d^2 x (a+b \arctan (c x))^2}{e^3}-\frac {d x^2 (a+b \arctan (c x))^2}{2 e^2}+\frac {x^3 (a+b \arctan (c x))^2}{3 e}+\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^4}+\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c e^3}-\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3 e}-\frac {d^3 (a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^4}-\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2 e^2}-\frac {i b d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^4}+\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c e^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3 e}+\frac {i b d^3 (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 )}{e^4}+\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^4}-\frac {b^2 d^3 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^4} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 38.26 (sec) , antiderivative size = 2062, normalized size of antiderivative = 3.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2062\)
default	\(\text {Expression too large to display}\)	\(2062\)
parts	\(\text {Expression too large to display}\)	\(2107\)

[In]

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

[Out]

1/c^4*(a^2*c^4/e^3*d^2*x-1/2*a^2*c^4/e^2*d*x^2+1/3*a^2*c^4/e*x^3-a^2*c^4*d^3/e^4*ln(c*e*x+c*d)+b^2*c*(1/3/e*(I
+c*x)+1/3*arctan(c*x)^2/e*c^3*x^3-I/e^4*c^3*d^3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+d^3*c^3/e^3/(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))-d^4*c^4/e^4/(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))+arctan(c*x)^2/e^3*c^3*d^2*x-1/2*arctan(c*x)^2/e^2*c^3*d*x^2
+1/2*I*d^3*c^3/e^3/(c*d-I*e)*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+1/2*I/e^4*c^3*d^3*Pi*csgn(
I/((1+I*c*x)^2/(c^2*x^2+1)+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+I
*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/2*I/e^4*c^3*d^3*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+I*c*x)
^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-I/e^3*c^2*d^2*arctan(c*x)^2-2*I/e^3*c^2*d^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)
^(1/2))+2/3*I/e*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2/3/e*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2
/3/e*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/3*arctan(c*x)*(c*x-I)^2/e+2/3*I/e*dilog(1-I*(1+I*c*x)/(
c^2*x^2+1)^(1/2))+1/3*I/e*arctan(c*x)^2+I*d^4*c^4/e^4/(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))+I*d^3*c^3/e^3/(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*I/e^4*c^3*d^3*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+I*c*x)^2/(c^
2*x^2+1)+1))^3*arctan(c*x)^2-1/2*I/e^4*c^3*d^3*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+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+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2+1/2/e^4*c^3*d^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-
2/3*arctan(c*x)*(c*x-I)*(I+c*x)/e+2/3*I*arctan(c*x)*(c*x-I)/e+1/e^2*c*d*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-1/2/e^2*
c*d*arctan(c*x)^2-arctan(c*x)^2*c^3*d^3/e^4*ln(c*e*x+c*d)-2*I/e^3*c^2*d^2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2
))-1/2*d^4*c^4/e^4/(c*d-I*e)*polylog(3,(I*e-c*d)/(c*d+I*e)*(1+I*c*x)^2/(c^2*x^2+1))+d^3*c^3/e^4*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)+2/e^3*c^2*d^2*arctan(c*x)*ln(1+I*(1+I*c*x
)/(c^2*x^2+1)^(1/2))+2/e^3*c^2*d^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+c*d*arctan(c*x)*(c*x-I)/e^2
)+2*a*b*c*(arctan(c*x)/e^3*c^3*d^2*x-1/2*arctan(c*x)/e^2*c^3*d*x^2+1/3*arctan(c*x)/e*c^3*x^3-arctan(c*x)*c^3*d
^3/e^4*ln(c*e*x+c*d)-1/e*(1/2/e^2*ln(c^2*d^2-2*c*d*(c*e*x+c*d)+e^2+(c*e*x+c*d)^2)*c^2*d^2+1/2/e*arctan(c*x)*c*
d-1/6*ln(c^2*d^2-2*c*d*(c*e*x+c*d)+e^2+(c*e*x+c*d)^2)-5/6/e^2*c*d*(c*e*x+c*d)+1/6/e^2*(c*e*x+c*d)^2-1/e^2*c^3*
d^3*(1/2*I*ln(c*e*x+c*d)*(-ln((I*e-e*c*x)/(c*d+I*e))+ln((I*e+e*c*x)/(I*e-c*d)))/e-1/2*I*(dilog((I*e-e*c*x)/(c*
d+I*e))-dilog((I*e+e*c*x)/(I*e-c*d)))/e))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/6*a^2*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 1/96*(96*e^3*integrate(1/48*(36*(b
^2*c^2*e^3*x^5 + b^2*e^3*x^3)*arctan(c*x)^2 + 3*(b^2*c^2*e^3*x^5 + b^2*e^3*x^3)*log(c^2*x^2 + 1)^2 + 4*(24*a*b
*c^2*e^3*x^5 - 2*b^2*c*e^3*x^4 - 3*b^2*c*d^2*e*x^2 - 6*b^2*c*d^3*x + (b^2*c*d*e^2 + 24*a*b*e^3)*x^3)*arctan(c*
x) + 2*(2*b^2*c^2*e^3*x^5 - b^2*c^2*d*e^2*x^4 + 3*b^2*c^2*d^2*e*x^3 + 6*b^2*c^2*d^3*x^2)*log(c^2*x^2 + 1))/(c^
2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d*e^3), x) + 4*(2*b^2*e^2*x^3 - 3*b^2*d*e*x^2 + 6*b^2*d^2*x)*arctan(c*x)^2
- (2*b^2*e^2*x^3 - 3*b^2*d*e*x^2 + 6*b^2*d^2*x)*log(c^2*x^2 + 1)^2)/e^3

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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