3.31 \(\int x^3 (d+i c d x)^4 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=238 \[ \frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} i b c^2 d^4 x^6+\frac{24 i b d^4 x^2}{35 c^2}-\frac{24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac{11 b d^4 x}{8 c^3}-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{9}{40} b c d^4 x^5-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4 \]

[Out]

(11*b*d^4*x)/(8*c^3) + (((24*I)/35)*b*d^4*x^2)/c^2 - (11*b*d^4*x^3)/(24*c) - ((12*I)/35)*b*d^4*x^4 + (9*b*c*d^
4*x^5)/40 + ((2*I)/21)*b*c^2*d^4*x^6 - (b*c^3*d^4*x^7)/56 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (d^4*x^4*(a + b*A
rcTan[c*x]))/4 + ((4*I)/5)*c*d^4*x^5*(a + b*ArcTan[c*x]) - c^2*d^4*x^6*(a + b*ArcTan[c*x]) - ((4*I)/7)*c^3*d^4
*x^7*(a + b*ArcTan[c*x]) + (c^4*d^4*x^8*(a + b*ArcTan[c*x]))/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4

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Rubi [A]  time = 0.214227, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {43, 4872, 12, 1802, 635, 203, 260} \[ \frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} i b c^2 d^4 x^6+\frac{24 i b d^4 x^2}{35 c^2}-\frac{24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac{11 b d^4 x}{8 c^3}-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{9}{40} b c d^4 x^5-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(11*b*d^4*x)/(8*c^3) + (((24*I)/35)*b*d^4*x^2)/c^2 - (11*b*d^4*x^3)/(24*c) - ((12*I)/35)*b*d^4*x^4 + (9*b*c*d^
4*x^5)/40 + ((2*I)/21)*b*c^2*d^4*x^6 - (b*c^3*d^4*x^7)/56 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (d^4*x^4*(a + b*A
rcTan[c*x]))/4 + ((4*I)/5)*c*d^4*x^5*(a + b*ArcTan[c*x]) - c^2*d^4*x^6*(a + b*ArcTan[c*x]) - ((4*I)/7)*c^3*d^4
*x^7*(a + b*ArcTan[c*x]) + (c^4*d^4*x^8*(a + b*ArcTan[c*x]))/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4872

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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]

Rubi steps

\begin{align*} \int x^3 (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d^4 x^4 \left (70+224 i c x-280 c^2 x^2-160 i c^3 x^3+35 c^4 x^4\right )}{280 \left (1+c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{280} \left (b c d^4\right ) \int \frac{x^4 \left (70+224 i c x-280 c^2 x^2-160 i c^3 x^3+35 c^4 x^4\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{280} \left (b c d^4\right ) \int \left (-\frac{385}{c^4}-\frac{384 i x}{c^3}+\frac{385 x^2}{c^2}+\frac{384 i x^3}{c}-315 x^4-160 i c x^5+35 c^2 x^6+\frac{385+384 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 i b d^4 x^2}{35 c^2}-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} i b c^2 d^4 x^6-\frac{1}{56} b c^3 d^4 x^7+\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (b d^4\right ) \int \frac{385+384 i c x}{1+c^2 x^2} \, dx}{280 c^3}\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 i b d^4 x^2}{35 c^2}-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} i b c^2 d^4 x^6-\frac{1}{56} b c^3 d^4 x^7+\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (11 b d^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{8 c^3}-\frac{\left (48 i b d^4\right ) \int \frac{x}{1+c^2 x^2} \, dx}{35 c^2}\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 i b d^4 x^2}{35 c^2}-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} i b c^2 d^4 x^6-\frac{1}{56} b c^3 d^4 x^7-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac{24 i b d^4 \log \left (1+c^2 x^2\right )}{35 c^4}\\ \end{align*}

Mathematica [A]  time = 0.155058, size = 290, normalized size = 1.22 \[ \frac{1}{8} a c^4 d^4 x^8-\frac{4}{7} i a c^3 d^4 x^7-a c^2 d^4 x^6+\frac{4}{5} i a c d^4 x^5+\frac{1}{4} a d^4 x^4-\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} i b c^2 d^4 x^6+\frac{24 i b d^4 x^2}{35 c^2}-\frac{24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac{1}{8} b c^4 d^4 x^8 \tan ^{-1}(c x)-\frac{4}{7} i b c^3 d^4 x^7 \tan ^{-1}(c x)-b c^2 d^4 x^6 \tan ^{-1}(c x)+\frac{11 b d^4 x}{8 c^3}-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{9}{40} b c d^4 x^5-\frac{11 b d^4 x^3}{24 c}+\frac{4}{5} i b c d^4 x^5 \tan ^{-1}(c x)+\frac{1}{4} b d^4 x^4 \tan ^{-1}(c x)-\frac{12}{35} i b d^4 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]

[Out]

(11*b*d^4*x)/(8*c^3) + (((24*I)/35)*b*d^4*x^2)/c^2 - (11*b*d^4*x^3)/(24*c) + (a*d^4*x^4)/4 - ((12*I)/35)*b*d^4
*x^4 + ((4*I)/5)*a*c*d^4*x^5 + (9*b*c*d^4*x^5)/40 - a*c^2*d^4*x^6 + ((2*I)/21)*b*c^2*d^4*x^6 - ((4*I)/7)*a*c^3
*d^4*x^7 - (b*c^3*d^4*x^7)/56 + (a*c^4*d^4*x^8)/8 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (b*d^4*x^4*ArcTan[c*x])/4
+ ((4*I)/5)*b*c*d^4*x^5*ArcTan[c*x] - b*c^2*d^4*x^6*ArcTan[c*x] - ((4*I)/7)*b*c^3*d^4*x^7*ArcTan[c*x] + (b*c^
4*d^4*x^8*ArcTan[c*x])/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4

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Maple [A]  time = 0.029, size = 249, normalized size = 1.1 \begin{align*}{\frac{{c}^{4}{d}^{4}a{x}^{8}}{8}}-{\frac{{\frac{24\,i}{35}}b{d}^{4}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{4}}}-{c}^{2}{d}^{4}a{x}^{6}-{\frac{12\,i}{35}}b{d}^{4}{x}^{4}+{\frac{{d}^{4}a{x}^{4}}{4}}+{\frac{{c}^{4}{d}^{4}b\arctan \left ( cx \right ){x}^{8}}{8}}+{\frac{2\,i}{21}}b{c}^{2}{d}^{4}{x}^{6}-{c}^{2}{d}^{4}b\arctan \left ( cx \right ){x}^{6}-{\frac{4\,i}{7}}{c}^{3}{d}^{4}b\arctan \left ( cx \right ){x}^{7}+{\frac{{d}^{4}b\arctan \left ( cx \right ){x}^{4}}{4}}+{\frac{11\,{d}^{4}bx}{8\,{c}^{3}}}-{\frac{b{c}^{3}{d}^{4}{x}^{7}}{56}}+{\frac{4\,i}{5}}c{d}^{4}a{x}^{5}+{\frac{9\,bc{d}^{4}{x}^{5}}{40}}+{\frac{4\,i}{5}}c{d}^{4}b\arctan \left ( cx \right ){x}^{5}-{\frac{11\,{d}^{4}b{x}^{3}}{24\,c}}+{\frac{{\frac{24\,i}{35}}b{d}^{4}{x}^{2}}{{c}^{2}}}-{\frac{4\,i}{7}}{c}^{3}{d}^{4}a{x}^{7}-{\frac{11\,b{d}^{4}\arctan \left ( cx \right ) }{8\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x)

[Out]

1/8*c^4*d^4*a*x^8-24/35*I*b*d^4*ln(c^2*x^2+1)/c^4-c^2*d^4*a*x^6-12/35*I*b*d^4*x^4+1/4*d^4*a*x^4+1/8*c^4*d^4*b*
arctan(c*x)*x^8+2/21*I*b*c^2*d^4*x^6-c^2*d^4*b*arctan(c*x)*x^6-4/7*I*c^3*d^4*b*arctan(c*x)*x^7+1/4*d^4*b*arcta
n(c*x)*x^4+11/8*b*d^4*x/c^3-1/56*b*c^3*d^4*x^7+4/5*I*c*d^4*a*x^5+9/40*b*c*d^4*x^5+4/5*I*c*d^4*b*arctan(c*x)*x^
5-11/24*b*d^4*x^3/c+24/35*I*b*d^4*x^2/c^2-4/7*I*c^3*d^4*a*x^7-11/8*b*d^4*arctan(c*x)/c^4

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Maxima [A]  time = 1.51219, size = 455, normalized size = 1.91 \begin{align*} \frac{1}{8} \, a c^{4} d^{4} x^{8} - \frac{4}{7} i \, a c^{3} d^{4} x^{7} - a c^{2} d^{4} x^{6} + \frac{4}{5} i \, a c d^{4} x^{5} + \frac{1}{840} \,{\left (105 \, x^{8} \arctan \left (c x\right ) - c{\left (\frac{15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac{105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} - \frac{1}{21} i \,{\left (12 \, x^{7} \arctan \left (c x\right ) - c{\left (\frac{2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac{6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac{1}{4} \, a d^{4} x^{4} - \frac{1}{15} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac{1}{5} i \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

1/8*a*c^4*d^4*x^8 - 4/7*I*a*c^3*d^4*x^7 - a*c^2*d^4*x^6 + 4/5*I*a*c*d^4*x^5 + 1/840*(105*x^8*arctan(c*x) - c*(
(15*c^6*x^7 - 21*c^4*x^5 + 35*c^2*x^3 - 105*x)/c^8 + 105*arctan(c*x)/c^9))*b*c^4*d^4 - 1/21*I*(12*x^7*arctan(c
*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*log(c^2*x^2 + 1)/c^8))*b*c^3*d^4 + 1/4*a*d^4*x^4 - 1/15*(15*x
^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*b*c^2*d^4 + 1/5*I*(4*x^5*arctan(
c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*b*c*d^4 + 1/12*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3
*x)/c^4 + 3*arctan(c*x)/c^5))*b*d^4

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Fricas [A]  time = 3.39207, size = 563, normalized size = 2.37 \begin{align*} \frac{210 \, a c^{8} d^{4} x^{8} +{\left (-960 i \, a - 30 \, b\right )} c^{7} d^{4} x^{7} - 80 \,{\left (21 \, a - 2 i \, b\right )} c^{6} d^{4} x^{6} +{\left (1344 i \, a + 378 \, b\right )} c^{5} d^{4} x^{5} + 12 \,{\left (35 \, a - 48 i \, b\right )} c^{4} d^{4} x^{4} - 770 \, b c^{3} d^{4} x^{3} + 1152 i \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 2307 i \, b d^{4} \log \left (\frac{c x + i}{c}\right ) + 3 i \, b d^{4} \log \left (\frac{c x - i}{c}\right ) +{\left (105 i \, b c^{8} d^{4} x^{8} + 480 \, b c^{7} d^{4} x^{7} - 840 i \, b c^{6} d^{4} x^{6} - 672 \, b c^{5} d^{4} x^{5} + 210 i \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{1680 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

1/1680*(210*a*c^8*d^4*x^8 + (-960*I*a - 30*b)*c^7*d^4*x^7 - 80*(21*a - 2*I*b)*c^6*d^4*x^6 + (1344*I*a + 378*b)
*c^5*d^4*x^5 + 12*(35*a - 48*I*b)*c^4*d^4*x^4 - 770*b*c^3*d^4*x^3 + 1152*I*b*c^2*d^4*x^2 + 2310*b*c*d^4*x - 23
07*I*b*d^4*log((c*x + I)/c) + 3*I*b*d^4*log((c*x - I)/c) + (105*I*b*c^8*d^4*x^8 + 480*b*c^7*d^4*x^7 - 840*I*b*
c^6*d^4*x^6 - 672*b*c^5*d^4*x^5 + 210*I*b*c^4*d^4*x^4)*log(-(c*x + I)/(c*x - I)))/c^4

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Sympy [A]  time = 4.11981, size = 355, normalized size = 1.49 \begin{align*} \frac{a c^{4} d^{4} x^{8}}{8} - \frac{11 b d^{4} x^{3}}{24 c} + \frac{24 i b d^{4} x^{2}}{35 c^{2}} + \frac{11 b d^{4} x}{8 c^{3}} + \frac{i b d^{4} \log{\left (x - \frac{i}{c} \right )}}{560 c^{4}} - \frac{769 i b d^{4} \log{\left (x + \frac{i}{c} \right )}}{560 c^{4}} + x^{7} \left (- \frac{4 i a c^{3} d^{4}}{7} - \frac{b c^{3} d^{4}}{56}\right ) + x^{6} \left (- a c^{2} d^{4} + \frac{2 i b c^{2} d^{4}}{21}\right ) + x^{5} \left (\frac{4 i a c d^{4}}{5} + \frac{9 b c d^{4}}{40}\right ) + x^{4} \left (\frac{a d^{4}}{4} - \frac{12 i b d^{4}}{35}\right ) + \left (- \frac{i b c^{4} d^{4} x^{8}}{16} - \frac{2 b c^{3} d^{4} x^{7}}{7} + \frac{i b c^{2} d^{4} x^{6}}{2} + \frac{2 b c d^{4} x^{5}}{5} - \frac{i b d^{4} x^{4}}{8}\right ) \log{\left (i c x + 1 \right )} + \left (\frac{i b c^{4} d^{4} x^{8}}{16} + \frac{2 b c^{3} d^{4} x^{7}}{7} - \frac{i b c^{2} d^{4} x^{6}}{2} - \frac{2 b c d^{4} x^{5}}{5} + \frac{i b d^{4} x^{4}}{8}\right ) \log{\left (- i c x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(d+I*c*d*x)**4*(a+b*atan(c*x)),x)

[Out]

a*c**4*d**4*x**8/8 - 11*b*d**4*x**3/(24*c) + 24*I*b*d**4*x**2/(35*c**2) + 11*b*d**4*x/(8*c**3) + I*b*d**4*log(
x - I/c)/(560*c**4) - 769*I*b*d**4*log(x + I/c)/(560*c**4) + x**7*(-4*I*a*c**3*d**4/7 - b*c**3*d**4/56) + x**6
*(-a*c**2*d**4 + 2*I*b*c**2*d**4/21) + x**5*(4*I*a*c*d**4/5 + 9*b*c*d**4/40) + x**4*(a*d**4/4 - 12*I*b*d**4/35
) + (-I*b*c**4*d**4*x**8/16 - 2*b*c**3*d**4*x**7/7 + I*b*c**2*d**4*x**6/2 + 2*b*c*d**4*x**5/5 - I*b*d**4*x**4/
8)*log(I*c*x + 1) + (I*b*c**4*d**4*x**8/16 + 2*b*c**3*d**4*x**7/7 - I*b*c**2*d**4*x**6/2 - 2*b*c*d**4*x**5/5 +
I*b*d**4*x**4/8)*log(-I*c*x + 1)

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Giac [A]  time = 1.16692, size = 354, normalized size = 1.49 \begin{align*} \frac{210 \, b c^{8} d^{4} x^{8} \arctan \left (c x\right ) + 210 \, a c^{8} d^{4} x^{8} - 960 \, b c^{7} d^{4} i x^{7} \arctan \left (c x\right ) - 960 \, a c^{7} d^{4} i x^{7} - 30 \, b c^{7} d^{4} x^{7} + 160 \, b c^{6} d^{4} i x^{6} - 1680 \, b c^{6} d^{4} x^{6} \arctan \left (c x\right ) - 1680 \, a c^{6} d^{4} x^{6} + 1344 \, b c^{5} d^{4} i x^{5} \arctan \left (c x\right ) + 1344 \, a c^{5} d^{4} i x^{5} + 378 \, b c^{5} d^{4} x^{5} - 576 \, b c^{4} d^{4} i x^{4} + 420 \, b c^{4} d^{4} x^{4} \arctan \left (c x\right ) + 420 \, a c^{4} d^{4} x^{4} - 770 \, b c^{3} d^{4} x^{3} + 1152 \, b c^{2} d^{4} i x^{2} + 2310 \, b c d^{4} x + 3 \, b d^{4} i \log \left (c i x + 1\right ) - 2307 \, b d^{4} i \log \left (-c i x + 1\right )}{1680 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

1/1680*(210*b*c^8*d^4*x^8*arctan(c*x) + 210*a*c^8*d^4*x^8 - 960*b*c^7*d^4*i*x^7*arctan(c*x) - 960*a*c^7*d^4*i*
x^7 - 30*b*c^7*d^4*x^7 + 160*b*c^6*d^4*i*x^6 - 1680*b*c^6*d^4*x^6*arctan(c*x) - 1680*a*c^6*d^4*x^6 + 1344*b*c^
5*d^4*i*x^5*arctan(c*x) + 1344*a*c^5*d^4*i*x^5 + 378*b*c^5*d^4*x^5 - 576*b*c^4*d^4*i*x^4 + 420*b*c^4*d^4*x^4*a
rctan(c*x) + 420*a*c^4*d^4*x^4 - 770*b*c^3*d^4*x^3 + 1152*b*c^2*d^4*i*x^2 + 2310*b*c*d^4*x + 3*b*d^4*i*log(c*i
*x + 1) - 2307*b*d^4*i*log(-c*i*x + 1))/c^4