\(\int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx\) [120]

Optimal result

Integrand size = 22, antiderivative size = 382 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=-3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \arctan (c x)}{4 c}-3 b^3 d^3 x \arctan (c x)-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {11 b^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c} \]

[Out]

-3*a*b^2*d^3*x+1/4*I*b^3*d^3*x-1/4*I*b^2*c*d^3*x^2*(a+b*arctan(c*x))-3*b^3*d^3*x*arctan(c*x)-1/4*I*b^3*d^3*arc
tan(c*x)/c+7*b*d^3*(a+b*arctan(c*x))^2/c-6*I*b^2*d^3*(a+b*arctan(c*x))*polylog(2,1-2/(1-I*c*x))/c+3/2*b*c*d^3*
x^2*(a+b*arctan(c*x))^2-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^3/c-21/4*I*b*d^3*x*(a+b*arctan(c*x))^2+6*b*d^3
*(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/c-11*I*b^2*d^3*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c+3/2*b^3*d^3*ln(c^2*x^2
+1)/c+1/4*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))^2+11/2*b^3*d^3*polylog(2,1-2/(1+I*c*x))/c+3*b^3*d^3*polylog(3,1-2/
(1-I*c*x))/c

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4974, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 266, 5004, 327, 209, 1600, 5112, 6745} \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=-\frac {6 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c}-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))-\frac {11 i b^2 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {6 b d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))^2}{c}-3 a b^2 d^3 x-\frac {i b^3 d^3 \arctan (c x)}{4 c}-3 b^3 d^3 x \arctan (c x)+\frac {3 b^3 d^3 \log \left (c^2 x^2+1\right )}{2 c}+\frac {11 b^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c}+\frac {1}{4} i b^3 d^3 x \]

[In]

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

[Out]

-3*a*b^2*d^3*x + (I/4)*b^3*d^3*x - ((I/4)*b^3*d^3*ArcTan[c*x])/c - 3*b^3*d^3*x*ArcTan[c*x] - (I/4)*b^2*c*d^3*x
^2*(a + b*ArcTan[c*x]) + (7*b*d^3*(a + b*ArcTan[c*x])^2)/c - ((21*I)/4)*b*d^3*x*(a + b*ArcTan[c*x])^2 + (3*b*c
*d^3*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/4)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x])^2 - ((I/4)*d^3*(1 + I*c*x)^4*(a +
b*ArcTan[c*x])^3)/c + (6*b*d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - ((11*I)*b^2*d^3*(a + b*ArcTan[c*x
])*Log[2/(1 + I*c*x)])/c + (3*b^3*d^3*Log[1 + c^2*x^2])/(2*c) - ((6*I)*b^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2,
1 - 2/(1 - I*c*x)])/c + (11*b^3*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d^3*PolyLog[3, 1 - 2/(1 - I*
c*x)])/c

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 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[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 {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {(3 i b) \int \left (-7 d^4 (a+b \arctan (c x))^2-4 i c d^4 x (a+b \arctan (c x))^2+c^2 d^4 x^2 (a+b \arctan (c x))^2-\frac {8 i \left (i d^4-c d^4 x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2}\right ) \, dx}{4 d} \\ & = -\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {(6 b) \int \frac {\left (i d^4-c d^4 x\right ) (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{d}-\frac {1}{4} \left (21 i b d^3\right ) \int (a+b \arctan (c x))^2 \, dx+\left (3 b c d^3\right ) \int x (a+b \arctan (c x))^2 \, dx+\frac {1}{4} \left (3 i b c^2 d^3\right ) \int x^2 (a+b \arctan (c x))^2 \, dx \\ & = -\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {(6 b) \int \frac {(a+b \arctan (c x))^2}{-\frac {i}{d^4}-\frac {c x}{d^4}} \, dx}{d}+\frac {1}{2} \left (21 i b^2 c d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{2} \left (i b^2 c^3 d^3\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {21 b d^3 (a+b \arctan (c x))^2}{4 c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {1}{2} \left (21 i b^2 d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx-\left (3 b^2 d^3\right ) \int (a+b \arctan (c x)) \, dx+\left (3 b^2 d^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx-\left (12 b^2 d^3\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\frac {1}{2} \left (i b^2 c d^3\right ) \int x (a+b \arctan (c x)) \, dx+\frac {1}{2} \left (i b^2 c d^3\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -3 a b^2 d^3 x-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {21 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{2} \left (i b^2 d^3\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx+\left (6 i b^3 d^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{2} \left (21 i b^3 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 b^3 d^3\right ) \int \arctan (c x) \, dx+\frac {1}{4} \left (i b^3 c^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx \\ & = -3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-3 b^3 d^3 x \arctan (c x)-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{4} \left (i b^3 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {1}{2} \left (i b^3 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {\left (21 b^3 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c}+\left (3 b^3 c d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \arctan (c x)}{4 c}-3 b^3 d^3 x \arctan (c x)-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {21 b^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c}+\frac {\left (b^3 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c} \\ & = -3 a b^2 d^3 x+\frac {1}{4} i b^3 d^3 x-\frac {i b^3 d^3 \arctan (c x)}{4 c}-3 b^3 d^3 x \arctan (c x)-\frac {1}{4} i b^2 c d^3 x^2 (a+b \arctan (c x))+\frac {7 b d^3 (a+b \arctan (c x))^2}{c}-\frac {21}{4} i b d^3 x (a+b \arctan (c x))^2+\frac {3}{2} b c d^3 x^2 (a+b \arctan (c x))^2+\frac {1}{4} i b c^2 d^3 x^3 (a+b \arctan (c x))^2-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^3}{4 c}+\frac {6 b d^3 (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{c}-\frac {11 i b^2 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac {6 i b^2 d^3 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}+\frac {11 b^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.81 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=-\frac {i d^3 \left (a b^2+4 i a^3 c x+21 a^2 b c x-12 i a b^2 c x-b^3 c x-6 a^3 c^2 x^2+6 i a^2 b c^2 x^2+a b^2 c^2 x^2-4 i a^3 c^3 x^3-a^2 b c^3 x^3+a^3 c^4 x^4-21 a^2 b \arctan (c x)+12 i a b^2 \arctan (c x)+b^3 \arctan (c x)+12 i a^2 b c x \arctan (c x)+42 a b^2 c x \arctan (c x)-12 i b^3 c x \arctan (c x)-18 a^2 b c^2 x^2 \arctan (c x)+12 i a b^2 c^2 x^2 \arctan (c x)+b^3 c^2 x^2 \arctan (c x)-12 i a^2 b c^3 x^3 \arctan (c x)-2 a b^2 c^3 x^3 \arctan (c x)+3 a^2 b c^4 x^4 \arctan (c x)+3 a b^2 \arctan (c x)^2-16 i b^3 \arctan (c x)^2+12 i a b^2 c x \arctan (c x)^2+21 b^3 c x \arctan (c x)^2-18 a b^2 c^2 x^2 \arctan (c x)^2+6 i b^3 c^2 x^2 \arctan (c x)^2-12 i a b^2 c^3 x^3 \arctan (c x)^2-b^3 c^3 x^3 \arctan (c x)^2+3 a b^2 c^4 x^4 \arctan (c x)^2+b^3 \arctan (c x)^3+4 i b^3 c x \arctan (c x)^3-6 b^3 c^2 x^2 \arctan (c x)^3-4 i b^3 c^3 x^3 \arctan (c x)^3+b^3 c^4 x^4 \arctan (c x)^3+48 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+44 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-12 i a^2 b \log \left (1+c^2 x^2\right )-22 a b^2 \log \left (1+c^2 x^2\right )+6 i b^3 \log \left (1+c^2 x^2\right )+2 b^2 (12 a-11 i b+12 b \arctan (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+12 i b^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{4 c} \]

[In]

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

[Out]

((-1/4*I)*d^3*(a*b^2 + (4*I)*a^3*c*x + 21*a^2*b*c*x - (12*I)*a*b^2*c*x - b^3*c*x - 6*a^3*c^2*x^2 + (6*I)*a^2*b
*c^2*x^2 + a*b^2*c^2*x^2 - (4*I)*a^3*c^3*x^3 - a^2*b*c^3*x^3 + a^3*c^4*x^4 - 21*a^2*b*ArcTan[c*x] + (12*I)*a*b
^2*ArcTan[c*x] + b^3*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] + 42*a*b^2*c*x*ArcTan[c*x] - (12*I)*b^3*c*x*Ar
cTan[c*x] - 18*a^2*b*c^2*x^2*ArcTan[c*x] + (12*I)*a*b^2*c^2*x^2*ArcTan[c*x] + b^3*c^2*x^2*ArcTan[c*x] - (12*I)
*a^2*b*c^3*x^3*ArcTan[c*x] - 2*a*b^2*c^3*x^3*ArcTan[c*x] + 3*a^2*b*c^4*x^4*ArcTan[c*x] + 3*a*b^2*ArcTan[c*x]^2
- (16*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c*x]^2 + 21*b^3*c*x*ArcTan[c*x]^2 - 18*a*b^2*c^2*x^2*Arc
Tan[c*x]^2 + (6*I)*b^3*c^2*x^2*ArcTan[c*x]^2 - (12*I)*a*b^2*c^3*x^3*ArcTan[c*x]^2 - b^3*c^3*x^3*ArcTan[c*x]^2
+ 3*a*b^2*c^4*x^4*ArcTan[c*x]^2 + b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 - 6*b^3*c^2*x^2*ArcTan[c*x]^
3 - (4*I)*b^3*c^3*x^3*ArcTan[c*x]^3 + b^3*c^4*x^4*ArcTan[c*x]^3 + (48*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*Ar
cTan[c*x])] + 44*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*Ar
cTan[c*x])] - (12*I)*a^2*b*Log[1 + c^2*x^2] - 22*a*b^2*Log[1 + c^2*x^2] + (6*I)*b^3*Log[1 + c^2*x^2] + 2*b^2*(
12*a - (11*I)*b + 12*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*PolyLog[3, -E^((2*I)*ArcTa
n[c*x])]))/c

Maple [C] (warning: unable to verify)

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

Time = 61.01 (sec) , antiderivative size = 1513, normalized size of antiderivative = 3.96

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1513\)
default	\(\text {Expression too large to display}\)	\(1513\)
parts	\(\text {Expression too large to display}\)	\(1521\)

[In]

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

[Out]

1/c*(-1/4*I*d^3*a^3*(1+I*c*x)^4+d^3*b^3*(-1/4*I*arctan(c*x)^3*c^4*x^4-arctan(c*x)^3*c^3*x^3+3/2*I*arctan(c*x)^
3*c^2*x^2+arctan(c*x)^3*c*x-1/4*I*arctan(c*x)^3+3/4*I*(1/3*I+1/3*c*x+1/3*c^3*x^3*arctan(c*x)^2-7*arctan(c*x)^2
*c*x-8*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2/3*arctan(c*x)*(c*x-I)*(c*x+I)+1/3*arctan(c*x)*(c*x-I)
^2-44/3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-44/3*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2
*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^3*arctan(c*x)^2-2*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)^2-2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2+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)^2-2*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+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)^2-4*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+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+4*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)^2-2*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))*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)^2-2*I*arctan(c*x)^2*c^2*x^2+4*I*ln(1+(1+I*c
*x)^2/(c^2*x^2+1))+16/3*I*arctan(c*x)^2-4*I*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+44/3*I*dilog(1+I*(1+I*c*x)/(c^
2*x^2+1)^(1/2))+44/3*I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-8*I*arctan(c*x)^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2)
)-8*I*ln(2)*arctan(c*x)^2+4*I*arctan(c*x)^2*ln(c^2*x^2+1)+14/3*I*arctan(c*x)*(c*x-I)))+3*a*d^3*b^2*(-1/4*I*arc
tan(c*x)^2*c^4*x^4-c^3*x^3*arctan(c*x)^2+3/2*I*arctan(c*x)^2*c^2*x^2+arctan(c*x)^2*c*x-1/4*I*arctan(c*x)^2+1/2
*I*(-7*c*x*arctan(c*x)+1/3*c^3*x^3*arctan(c*x)+4*I*arctan(c*x)*ln(c^2*x^2+1)-2*I*arctan(c*x)*c^2*x^2+4*arctan(
c*x)^2-2*ln(c*x-I)*ln(c^2*x^2+1)+2*ln(c*x+I)*ln(c^2*x^2+1)+2*ln(c*x-I)*ln(-1/2*I*(c*x+I))+ln(c*x-I)^2-ln(c*x+I
)^2-2*ln(c*x+I)*ln(1/2*I*(c*x-I))+2*dilog(-1/2*I*(c*x+I))-2*dilog(1/2*I*(c*x-I))-2*I*arctan(c*x)-1/6*c^2*x^2+1
1/3*ln(c^2*x^2+1)+2*I*c*x))+3*d^3*a^2*b*(-1/4*I*arctan(c*x)*c^4*x^4-c^3*x^3*arctan(c*x)+3/2*I*arctan(c*x)*c^2*
x^2+c*x*arctan(c*x)-1/4*I*arctan(c*x)+1/4*I*(-7*c*x+1/3*c^3*x^3-2*I*c^2*x^2+4*I*ln(c^2*x^2+1)+8*arctan(c*x))))

Fricas [F]

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

[In]

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

[Out]

-1/32*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^2 + 4*I*b^3*d^3*x)*log(-(c*x + I)/(c*x - I))^3 +
integral(1/16*(-16*I*a^3*c^5*d^3*x^5 - 48*a^3*c^4*d^3*x^4 + 32*I*a^3*c^3*d^3*x^3 - 32*a^3*c^2*d^3*x^2 + 48*I*a
^3*c*d^3*x + 16*a^3*d^3 - 3*(-4*I*a*b^2*c^5*d^3*x^5 - (12*a*b^2 - I*b^3)*c^4*d^3*x^4 + 4*(2*I*a*b^2 + b^3)*c^3
*d^3*x^3 - 2*(4*a*b^2 + 3*I*b^3)*c^2*d^3*x^2 + 4*a*b^2*d^3 + 4*(3*I*a*b^2 - b^3)*c*d^3*x)*log(-(c*x + I)/(c*x
- I))^2 + 24*(a^2*b*c^5*d^3*x^5 - 3*I*a^2*b*c^4*d^3*x^4 - 2*a^2*b*c^3*d^3*x^3 - 2*I*a^2*b*c^2*d^3*x^2 - 3*a^2*
b*c*d^3*x + I*a^2*b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/4*I*a^3*c^3*d^3*x^4 - 24*b^3*c^5*d^3*integrate(1/128*x^5*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) +
2*b^3*c^5*d^3*integrate(1/128*x^5*log(c^2*x^2 + 1)^3/(c^2*x^2 + 1), x) - 12*b^3*c^5*d^3*integrate(1/128*x^5*a
rctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c^5*d^3*integrate(1/128*x^5*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - a^3*
c^2*d^3*x^3 - 336*b^3*c^4*d^3*integrate(1/128*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) - 36*b^3*c^4*d^3*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^4*d^3*integrate(1/128*x^4*arctan(c*x)
^2/(c^2*x^2 + 1), x) - 60*b^3*c^4*d^3*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 1/4
*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a^2*b*c^3*d^3 + 48*b^3*c^3*d^3*integrate(
1/128*x^3*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 4*b^3*c^3*d^3*integrate(1/128*x^3*log(c^2*x^2 + 1
)^3/(c^2*x^2 + 1), x) + 120*b^3*c^3*d^3*integrate(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) - 30*b^3*c^3*d^3*i
ntegrate(1/128*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 3/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 +
1)/c^4))*a^2*b*c^2*d^3 + 3/2*I*a^3*c*d^3*x^2 + 7/32*b^3*d^3*arctan(c*x)^4/c - 224*b^3*c^2*d^3*integrate(1/128*
x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) - 24*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) - 768*a*b^2*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 120*b^3*c^2*d^3*integrat
e(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 9/2*I*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c
^3))*a^2*b*c*d^3 + a*b^2*d^3*arctan(c*x)^3/c + 72*b^3*c*d^3*integrate(1/128*x*arctan(c*x)^2*log(c^2*x^2 + 1)/(
c^2*x^2 + 1), x) - 6*b^3*c*d^3*integrate(1/128*x*log(c^2*x^2 + 1)^3/(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)
+ a^3*d^3*x + 12*b^3*d^3*integrate(1/128*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^3/c - 1/32*(I*b^3*c^3*d^3*x^4 + 4*b^3*c^2*d^3*x^3 - 6*I*b^3*c*d^3*x^2 - 4*b^
3*d^3*x)*arctan(c*x)^3 + 3/64*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^2 + 4*I*b^3*d^3*x)*arctan
(c*x)^2*log(c^2*x^2 + 1) - 3/128*(-I*b^3*c^3*d^3*x^4 - 4*b^3*c^2*d^3*x^3 + 6*I*b^3*c*d^3*x^2 + 4*b^3*d^3*x)*ar
ctan(c*x)*log(c^2*x^2 + 1)^2 - 1/256*(b^3*c^3*d^3*x^4 - 4*I*b^3*c^2*d^3*x^3 - 6*b^3*c*d^3*x^2 + 4*I*b^3*d^3*x)
*log(c^2*x^2 + 1)^3 - I*integrate(1/128*(112*(b^3*c^5*d^3*x^5 - 2*b^3*c^3*d^3*x^3 - 3*b^3*c*d^3*x)*arctan(c*x)
^3 + 2*(3*b^3*c^4*d^3*x^4 + 2*b^3*c^2*d^3*x^2 - b^3*d^3)*log(c^2*x^2 + 1)^3 + 12*(32*a*b^2*c^5*d^3*x^5 - 5*b^3
*c^4*d^3*x^4 - 64*a*b^2*c^3*d^3*x^3 + 10*b^3*c^2*d^3*x^2 - 96*a*b^2*c*d^3*x)*arctan(c*x)^2 + 3*(5*b^3*c^4*d^3*
x^4 - 10*b^3*c^2*d^3*x^2 + 4*(b^3*c^5*d^3*x^5 - 2*b^3*c^3*d^3*x^3 - 3*b^3*c*d^3*x)*arctan(c*x))*log(c^2*x^2 +
1)^2 - 12*(2*(3*b^3*c^4*d^3*x^4 + 2*b^3*c^2*d^3*x^2 - b^3*d^3)*arctan(c*x)^2 - (b^3*c^5*d^3*x^5 - 10*b^3*c^3*d
^3*x^3 + 4*b^3*c*d^3*x)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]

[In]

int((a + b*atan(c*x))^3*(d + c*d*x*1i)^3,x)

[Out]

int((a + b*atan(c*x))^3*(d + c*d*x*1i)^3, x)