\(\int (d+e x)^3 (a+b \arctan (c x))^3 \, dx\) [15]

Optimal result

Integrand size = 18, antiderivative size = 652 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}+\frac {i b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {3 b^3 d (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \]

[Out]

3*a*b^2*d*e^2*x/c^2-1/4*b^3*e^3*x/c^3+1/4*b^3*e^3*arctan(c*x)/c^4+3*b^3*d*e^2*x*arctan(c*x)/c^2+1/4*b^2*e^3*x^
2*(a+b*arctan(c*x))/c^2-3/2*b*d*e^2*(a+b*arctan(c*x))^2/c^3-3/4*I*b*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))^2/c^4+
1/4*I*b*e^3*(a+b*arctan(c*x))^2/c^4-3/4*b*e*(6*c^2*d^2-e^2)*x*(a+b*arctan(c*x))^2/c^3-3/2*b*d*e^2*x^2*(a+b*arc
tan(c*x))^2/c-1/4*b*e^3*x^3*(a+b*arctan(c*x))^2/c-3/4*I*b^3*e*(6*c^2*d^2-e^2)*polylog(2,1-2/(1+I*c*x))/c^4-1/4
*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan(c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b*arctan(c*x))^3/e+1/2*b^2*e^3*(a+b*ar
ctan(c*x))*ln(2/(1+I*c*x))/c^4-3/2*b^2*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^4+3*b*d*(c*d-e)*(
c*d+e)*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/c^3-3/2*b^3*d*e^2*ln(c^2*x^2+1)/c^3+3*I*b^2*d*(c*d-e)*(c*d+e)*(a+b*
arctan(c*x))*polylog(2,1-2/(1+I*c*x))/c^3+I*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^3/c^3+1/4*I*b^3*e^3*polylog(2,
1-2/(1+I*c*x))/c^4+3/2*b^3*d*(c*d-e)*(c*d+e)*polylog(3,1-2/(1+I*c*x))/c^3

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4974, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 327, 209, 5104, 5114, 6745} \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {b^2 e^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{2 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^3}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{2 c^4}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}+\frac {3 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^3}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}-\frac {3 b e x \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {3 a b^2 d e^2 x}{c^2}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {i b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{4 c^4}+\frac {3 b^3 d (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^3}-\frac {b^3 e^3 x}{4 c^3}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{4 c^4}-\frac {3 b^3 d e^2 \log \left (c^2 x^2+1\right )}{2 c^3} \]

[In]

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

[Out]

(3*a*b^2*d*e^2*x)/c^2 - (b^3*e^3*x)/(4*c^3) + (b^3*e^3*ArcTan[c*x])/(4*c^4) + (3*b^3*d*e^2*x*ArcTan[c*x])/c^2
+ (b^2*e^3*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (3*b*d*e^2*(a + b*ArcTan[c*x])^2)/(2*c^3) + ((I/4)*b*e^3*(a + b*
ArcTan[c*x])^2)/c^4 - (((3*I)/4)*b*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^4 - (3*b*e*(6*c^2*d^2 - e^2)*x
*(a + b*ArcTan[c*x])^2)/(4*c^3) - (3*b*d*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) - (b*e^3*x^3*(a + b*ArcTan[c*x])
^2)/(4*c) + (I*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/c^3 - ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcT
an[c*x])^3)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x])^3)/(4*e) + (b^2*e^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*
c*x)])/(2*c^4) - (3*b^2*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(2*c^4) + (3*b*d*(c*d - e)
*(c*d + e)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (3*b^3*d*e^2*Log[1 + c^2*x^2])/(2*c^3) + ((I/4)*b^3
*e^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4 - (((3*I)/4)*b^3*e*(6*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^
4 + ((3*I)*b^2*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (3*b^3*d*(c*d -
e)*(c*d + e)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)

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 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 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 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 {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {(3 b c) \int \left (\frac {e^2 \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{c^4}+\frac {4 d e^3 x (a+b \arctan (c x))^2}{c^2}+\frac {e^4 x^2 (a+b \arctan (c x))^2}{c^2}+\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) (a+b \arctan (c x))^2}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 e} \\ & = \frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {(3 b) \int \frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{4 c^3 e}-\frac {\left (3 b d e^2\right ) \int x (a+b \arctan (c x))^2 \, dx}{c}-\frac {\left (3 b e^3\right ) \int x^2 (a+b \arctan (c x))^2 \, dx}{4 c}-\frac {\left (3 b e \left (6 c^2 d^2-e^2\right )\right ) \int (a+b \arctan (c x))^2 \, dx}{4 c^3} \\ & = -\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {(3 b) \int \left (\frac {c^4 d^4 \left (1+\frac {-6 c^2 d^2 e^2+e^4}{c^4 d^4}\right ) (a+b \arctan (c x))^2}{1+c^2 x^2}+\frac {4 c^2 d (c d-e) e (c d+e) x (a+b \arctan (c x))^2}{1+c^2 x^2}\right ) \, dx}{4 c^3 e}+\left (3 b^2 d e^2\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {1}{2} \left (b^2 e^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {\left (3 b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 c^2} \\ & = -\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {\left (3 b^2 d e^2\right ) \int (a+b \arctan (c x)) \, dx}{c^2}-\frac {\left (3 b^2 d e^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c^2}+\frac {\left (b^2 e^3\right ) \int x (a+b \arctan (c x)) \, dx}{2 c^2}-\frac {\left (b^2 e^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 c^2}-\frac {(3 b d (c d-e) (c d+e)) \int \frac {x (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{c}-\frac {\left (3 b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{2 c^3}-\frac {\left (3 b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )\right ) \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{4 c^3 e} \\ & = \frac {3 a b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {\left (3 b^3 d e^2\right ) \int \arctan (c x) \, dx}{c^2}+\frac {\left (b^2 e^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{2 c^3}-\frac {\left (b^3 e^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{4 c}+\frac {(3 b d (c d-e) (c d+e)) \int \frac {(a+b \arctan (c x))^2}{i-c x} \, dx}{c^2}+\frac {\left (3 b^3 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3} \\ & = \frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {\left (3 b^3 d e^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c}+\frac {\left (b^3 e^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3}-\frac {\left (b^3 e^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}-\frac {\left (6 b^2 d (c d-e) (c d+e)\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}-\frac {\left (3 i b^3 e \left (6 c^2 d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c^4} \\ & = \frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {\left (i b^3 e^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c^4}-\frac {\left (3 i b^3 d (c d-e) (c d+e)\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2} \\ & = \frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}+\frac {i b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {3 b^3 d (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.01 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.31 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {a^2 c \left (4 a c^3 d^3+3 b e \left (-6 c^2 d^2+e^2\right )\right ) x+6 a^2 c^3 d e (a c d-b e) x^2+a^2 c^3 e^2 (4 a c d-b e) x^3+a^3 c^4 e^3 x^4+3 a^2 b \left (6 c^2 d^2 e-e^3\right ) \arctan (c x)+3 a^2 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \arctan (c x)+a b^2 e^3 \left (1+c^2 x^2+\left (6 c x-2 c^3 x^3\right ) \arctan (c x)+3 \left (-1+c^4 x^4\right ) \arctan (c x)^2-4 \log \left (1+c^2 x^2\right )\right )-6 a^2 b c d \left (c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )+18 a b^2 c^2 d^2 e \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )+12 a b^2 c^3 d^3 \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+12 a b^2 c d e^2 \left (c x+\left (i+c^3 x^3\right ) \arctan (c x)^2-\arctan (c x) \left (1+c^2 x^2+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 b^3 c^2 d^2 e \left (\arctan (c x) \left ((3 i-3 c x) \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-6 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 e^3 \left (-c x-\left (4 i-3 c x+c^3 x^3\right ) \arctan (c x)^2+\left (-1+c^4 x^4\right ) \arctan (c x)^3+\arctan (c x) \left (1+c^2 x^2+8 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-4 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+2 b^3 c d e^2 \left (6 c x \arctan (c x)-3 \arctan (c x)^2-3 c^2 x^2 \arctan (c x)^2+2 i \arctan (c x)^3+2 c^3 x^3 \arctan (c x)^3-6 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 \log \left (1+c^2 x^2\right )+6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )+2 b^3 c^3 d^3 \left (2 \arctan (c x)^2 \left ((-i+c x) \arctan (c x)+3 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{4 c^4} \]

[In]

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

[Out]

(a^2*c*(4*a*c^3*d^3 + 3*b*e*(-6*c^2*d^2 + e^2))*x + 6*a^2*c^3*d*e*(a*c*d - b*e)*x^2 + a^2*c^3*e^2*(4*a*c*d - b
*e)*x^3 + a^3*c^4*e^3*x^4 + 3*a^2*b*(6*c^2*d^2*e - e^3)*ArcTan[c*x] + 3*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e
^2*x^2 + e^3*x^3)*ArcTan[c*x] + a*b^2*e^3*(1 + c^2*x^2 + (6*c*x - 2*c^3*x^3)*ArcTan[c*x] + 3*(-1 + c^4*x^4)*Ar
cTan[c*x]^2 - 4*Log[1 + c^2*x^2]) - 6*a^2*b*c*d*(c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c^2*d^2*e*(-2*c*x*
ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 12*a*b^2*c^3*d^3*(ArcTan[c*x]*((-I + c*x)*ArcT
an[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 12*a*b^2*c*d*e^2*(c*x +
(I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^(
(2*I)*ArcTan[c*x])]) + 6*b^3*c^2*d^2*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 -
6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*e^3*(-(c*x) - (4*I - 3*c*
x + c^3*x^3)*ArcTan[c*x]^2 + (-1 + c^4*x^4)*ArcTan[c*x]^3 + ArcTan[c*x]*(1 + c^2*x^2 + 8*Log[1 + E^((2*I)*ArcT
an[c*x])]) - (4*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c*d*e^2*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 -
3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*Arc
Tan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I
)*ArcTan[c*x])]) + 2*b^3*c^3*d^3*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2*I)*ArcTan[c*x])])
- (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(4*c^4)

Maple [C] (warning: unable to verify)

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

Time = 93.44 (sec) , antiderivative size = 3122, normalized size of antiderivative = 4.79

method	result	size
parts	\(\text {Expression too large to display}\)	\(3122\)
derivativedivides	\(\text {Expression too large to display}\)	\(3153\)
default	\(\text {Expression too large to display}\)	\(3153\)

[In]

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

[Out]

1/4*a^3*(e*x+d)^4/e+b^3/c*(1/4*c*e^3*arctan(c*x)^3*x^4+c*e^2*arctan(c*x)^3*x^3*d+3/2*c*e*arctan(c*x)^3*x^2*d^2
+arctan(c*x)^3*c*x*d^3+1/4*c/e*arctan(c*x)^3*d^4-3/4/c^3/e*(2*I*e^3*c*d*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*arctan(c*x)^2-I*e*c^3*d^3*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)*arctan(c*x)^2-2*I*e*c^3*d^3*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*arctan(c*x)^2+I*e*c^3*d^3*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))*arctan(c*x)^2-I*e*c^3*d^3*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*arctan(c*x)^2+2*I*e*c^3*d^3*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*arctan(c*x)^2-I*e*c^3*d^3*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*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))^2*cs
gn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan(c*x)^2-2*I*e^3*c*d*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*arctan(c*x)^2+I*e^3*c*d*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*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*c
sgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2-I*e^3*c*d*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))*arctan(c*x)^2+1/3*e^4*arctan(c*x)*(c*x-I)^2+arctan(c*x)^3*c^
4*d^4+4/3*I*e^4*arctan(c*x)^2-8/3*e^4*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-8/3*e^4*arctan(c*x)*ln(1
-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+8/3*I*e^4*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+8/3*I*e^4*dilog(1-I*(1+I*c*x)
/(c^2*x^2+1)^(1/2))+arctan(c*x)^3*e^4-1/9*(6*c^4*d^4-36*c^2*d^2*e^2+6*e^4)*arctan(c*x)^3-I*e^3*c*d*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*arctan(c*x)^2-I*e^3*c*d*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2
+1))^3*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*arctan(c*x)^2+I*e*c^3*d^3*Pi*csgn(I*
(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2+I*e*c^3*d^3*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*arctan(c*x)^2-I*e*c^3*d^3*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*arctan(c*x)^2+2*arctan(c*x)^2*l
n(c^2*x^2+1)*c^3*d^3*e-2*arctan(c*x)^2*ln(c^2*x^2+1)*c*d*e^3+12*e^2*d^2*c^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*
x^2+1)^(1/2))+12*e^2*d^2*c^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-4*e*ln(2)*c^3*d^3*arctan(c*x)^2+4
*e^3*ln(2)*c*d*arctan(c*x)^2+1/3*arctan(c*x)^2*e^4*c^3*x^3-arctan(c*x)^2*e^4*c*x-6*arctan(c*x)^3*c^2*d^2*e^2-I
*e^3*c*d*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)*arctan(c*x)^2+I*e*c^3*d^3*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)*arctan(c*x)^2-4*e^3*c*d*a
rctan(c*x)*(c*x-I)-4*e*d^3*c^3*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)^2+4*e^3*d*c*ln((1+I*c*x)/(c^2*x^2+1
)^(1/2))*arctan(c*x)^2-12*I*e^2*d^2*c^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-12*I*e^2*d^2*c^2*dilog(1-I*(1+I
*c*x)/(c^2*x^2+1)^(1/2))-6*I*e^2*d^2*c^2*arctan(c*x)^2+4/3*I*e*d^3*c^3*arctan(c*x)^3-4/3*I*e^3*d*c*arctan(c*x)
^3+1/3*e^4*(c*x+I)-2*e*c^3*d^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+2*e^3*c*d*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1
))-4*e^3*c*d*ln(1+(1+I*c*x)^2/(c^2*x^2+1))+2*e^3*c*d*arctan(c*x)^2+2/3*I*e^4*arctan(c*x)*(c*x-I)-2/3*e^4*arcta
n(c*x)*(c*x-I)*(c*x+I)+6*arctan(c*x)^2*c^3*d^2*e^2*x+2*arctan(c*x)^2*e^3*c^3*d*x^2+4*I*e*c^3*d^3*arctan(c*x)*p
olylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-4*I*e^3*c*d*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))))+3*a*b^2/c*(1/
4*c*e^3*arctan(c*x)^2*x^4+c*e^2*arctan(c*x)^2*x^3*d+3/2*c*e*arctan(c*x)^2*x^2*d^2+arctan(c*x)^2*c*x*d^3+1/4*c/
e*arctan(c*x)^2*d^4-1/2/c^3/e*(6*arctan(c*x)*c^3*d^2*e^2*x+2*arctan(c*x)*e^3*c^3*d*x^2+1/3*arctan(c*x)*e^4*c^3
*x^3-arctan(c*x)*e^4*c*x+2*arctan(c*x)*ln(c^2*x^2+1)*c^3*d^3*e-2*arctan(c*x)*ln(c^2*x^2+1)*c*d*e^3+arctan(c*x)
^2*c^4*d^4-6*arctan(c*x)^2*c^2*d^2*e^2+arctan(c*x)^2*e^4-1/12*(6*c^4*d^4-36*c^2*d^2*e^2+6*e^4)*arctan(c*x)^2-1
/3*e^2*(6*c^2*d*e*x+1/2*c^2*e^2*x^2+1/2*(18*c^2*d^2-4*e^2)*ln(c^2*x^2+1)-6*e*arctan(c*x)*c*d)-2*c*d*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*(1/
4*c*e^3*arctan(c*x)*x^4+c*e^2*arctan(c*x)*x^3*d+3/2*c*e*arctan(c*x)*x^2*d^2+arctan(c*x)*c*x*d^3+1/4*c/e*arctan
(c*x)*d^4-1/4/c^3/e*(6*c^3*d^2*e^2*x+2*e^3*c^3*d*x^2+1/3*e^4*c^3*x^3-c*e^4*x+1/2*(4*c^3*d^3*e-4*c*d*e^3)*ln(c^
2*x^2+1)+(c^4*d^4-6*c^2*d^2*e^2+e^4)*arctan(c*x)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 7/32*b^3*d^3*arctan(c*x)^4/c + 112*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*
x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) +
384*a*b^2*c^2*e^3*integrate(1/128*x^5*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*c^2*d*e^2*integrate(1/128*x^4
*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 +
1), x) + 36*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^
2*d*e^2*integrate(1/128*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x
)^3/(c^2*x^2 + 1), x) + 48*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) +
36*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d^2*e*i
ntegrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 112*b^3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^3/(c^2*x^
2 + 1), x) + 72*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 12*b^3*c^2*
d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 384*a*b^2*c^2*d^3*integrate(1/128*x
^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 48*b^3*c^2*d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2
+ 1), x) + 3/2*a^3*d^2*e*x^2 + a*b^2*d^3*arctan(c*x)^3/c - 12*b^3*c*e^3*integrate(1/128*x^4*arctan(c*x)^2/(c^2
*x^2 + 1), x) + 3*b^3*c*e^3*integrate(1/128*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 48*b^3*c*d*e^2*integrat
e(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3*c*d*e^2*integrate(1/128*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 +
1), x) - 72*b^3*c*d^2*e*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 18*b^3*c*d^2*e*integrate(1/128*
x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 48*b^3*c*d^3*integrate(1/128*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12
*b^3*c*d^3*integrate(1/128*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 9/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c
*x)/c^3))*a^2*b*d^2*e + 3/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a^2*b*d*e^2 + 1/4*(3*x^4*
arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a^2*b*e^3 + a^3*d^3*x + 112*b^3*e^3*integrate(1/128
*x^3*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*e^3*integrate(1/128*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2
+ 1), x) + 384*a*b^2*e^3*integrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*d*e^2*integrate(1/128*x
^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 36*b^3*d*e^2*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2
+ 1), x) + 1152*a*b^2*d*e^2*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336*b^3*d^2*e*integrate(1/12
8*x*arctan(c*x)^3/(c^2*x^2 + 1), x) + 36*b^3*d^2*e*integrate(1/128*x*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 +
1), x) + 1152*a*b^2*d^2*e*integrate(1/128*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^3*d^3*integrate(1/128*arct
an(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^3/c + 1/32*(
b^3*e^3*x^4 + 4*b^3*d*e^2*x^3 + 6*b^3*d^2*e*x^2 + 4*b^3*d^3*x)*arctan(c*x)^3 - 3/128*(b^3*e^3*x^4 + 4*b^3*d*e^
2*x^3 + 6*b^3*d^2*e*x^2 + 4*b^3*d^3*x)*arctan(c*x)*log(c^2*x^2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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