\(\int (d+e x)^2 (a+b \arctan (c x))^3 \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 411 \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \arctan (c x)}{c^2}-\frac {3 i b d e (a+b \arctan (c x))^2}{c^2}-\frac {b e^2 (a+b \arctan (c x))^2}{2 c^3}-\frac {3 b d e x (a+b \arctan (c x))^2}{c}-\frac {b e^2 x^2 (a+b \arctan (c x))^2}{2 c}+\frac {i \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^3}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) (a+b \arctan (c x))^3}{3 e}+\frac {(d+e x)^3 (a+b \arctan (c x))^3}{3 e}-\frac {6 b^2 d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {b \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^3 e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac {3 i b^3 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {4974, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 5104, 5114, 6745} \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=-\frac {6 b^2 d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^3}-\frac {b e^2 (a+b \arctan (c x))^2}{2 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) (a+b \arctan (c x))^3}{3 e}-\frac {3 i b d e (a+b \arctan (c x))^2}{c^2}+\frac {i \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^3}{3 c^3}+\frac {b \left (3 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^3}-\frac {3 b d e x (a+b \arctan (c x))^2}{c}+\frac {(d+e x)^3 (a+b \arctan (c x))^3}{3 e}-\frac {b e^2 x^2 (a+b \arctan (c x))^2}{2 c}+\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \arctan (c x)}{c^2}-\frac {3 i b^3 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2}+\frac {b^3 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^3}-\frac {b^3 e^2 \log \left (c^2 x^2+1\right )}{2 c^3} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.51 \[ \int (d+e x)^2 (a+b \arctan (c x))^3 \, dx=\frac {6 a^2 c^2 d (a c d-3 b e) x+3 a^2 c^2 e (2 a c d-b e) x^2+2 a^3 c^3 e^2 x^3+18 a^2 b c d e \arctan (c x)+6 a^2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \arctan (c x)-3 a^2 b \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )+18 a b^2 c d e \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )+18 a b^2 c^2 d^2 \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 a b^2 e^2 \left (c x+\left (i+c^3 x^3\right ) \arctan (c x)^2-\arctan (c x) \left (1+c^2 x^2+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+6 b^3 c d e \left (\arctan (c x) \left ((3 i-3 c x) \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-6 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 e^2 \left (6 c x \arctan (c x)-3 \arctan (c x)^2-3 c^2 x^2 \arctan (c x)^2+2 i \arctan (c x)^3+2 c^3 x^3 \arctan (c x)^3-6 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-3 \log \left (1+c^2 x^2\right )+6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )+3 b^3 c^2 d^2 \left (2 \arctan (c x)^2 \left ((-i+c x) \arctan (c x)+3 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{6 c^3} \]

[In]

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

[Out]

(6*a^2*c^2*d*(a*c*d - 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d - b*e)*x^2 + 2*a^3*c^3*e^2*x^3 + 18*a^2*b*c*d*e*ArcTan[c
*x] + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x] - 3*a^2*b*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*
a*b^2*c*d*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 18*a*b^2*c^2*d^2*(ArcTan[c
*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*a*
b^2*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*d*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcT
an[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^2*(6*c*x*Arc
Tan[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*ArcTa
n[c*x]^2*Log[1 + E^((2*I)*ArcTan[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])]) + 3*b^3*c^2*d^2*(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)*A
rcTan[c*x])]))/(6*c^3)

Maple [C] (warning: unable to verify)

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

Time = 32.97 (sec) , antiderivative size = 2633, normalized size of antiderivative = 6.41

method	result	size
parts	\(\text {Expression too large to display}\)	\(2633\)
derivativedivides	\(\text {Expression too large to display}\)	\(2647\)
default	\(\text {Expression too large to display}\)	\(2647\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/3*a^3*e^2*x^3 + 7/32*b^3*d^2*arctan(c*x)^4/c + 28*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)^3/(c^2*x^2 + 1)
, x) + 3*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*e^2*in
tegrate(1/32*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 56*b^3*c^2*d*e*integrate(1/32*x^3*arctan(c*x)^3/(c^2*x^2 +
1), x) + 4*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 6*b^3*c^2*d*e*integ
rate(1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2*d*e*integrate(1/32*x^3*arctan(c
*x)^2/(c^2*x^2 + 1), x) + 28*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*d*e*i
ntegrate(1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x
)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*d^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) +
12*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + a^3*d*e*x^2 + a*b^2*d^2*ar
ctan(c*x)^3/c - 4*b^3*c*e^2*integrate(1/32*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^3*c*e^2*integrate(1/32*x^3*
log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 12*b^3*c*d*e*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3
*c*d*e*integrate(1/32*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 12*b^3*c*d^2*integrate(1/32*x*arctan(c*x)^2/(
c^2*x^2 + 1), x) + 3*b^3*c*d^2*integrate(1/32*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3*(x^2*arctan(c*x) - c*
(x/c^2 - arctan(c*x)/c^3))*a^2*b*d*e + 1/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a^2*b*e^2
+ a^3*d^2*x + 28*b^3*e^2*integrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 3*b^3*e^2*integrate(1/32*x^2*arc
tan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*e^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x
) + 56*b^3*d*e*integrate(1/32*x*arctan(c*x)^3/(c^2*x^2 + 1), x) + 6*b^3*d*e*integrate(1/32*x*arctan(c*x)*log(c
^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*d*e*integrate(1/32*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*d^2*i
ntegrate(1/32*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a^
2*b*d^2/c + 1/24*(b^3*e^2*x^3 + 3*b^3*d*e*x^2 + 3*b^3*d^2*x)*arctan(c*x)^3 - 1/32*(b^3*e^2*x^3 + 3*b^3*d*e*x^2
+ 3*b^3*d^2*x)*arctan(c*x)*log(c^2*x^2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[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)