\(\int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx\) [31]

Optimal result

Integrand size = 23, antiderivative size = 238 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {11 b d^4 x}{8 c^3}+\frac {24 i b d^4 x^2}{35 c^2}-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} i b c^2 d^4 x^6-\frac {1}{56} b c^3 d^4 x^7-\frac {11 b d^4 \arctan (c x)}{8 c^4}+\frac {1}{4} d^4 x^4 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\frac {24 i b d^4 \log \left (1+c^2 x^2\right )}{35 c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {45, 4992, 12, 1816, 649, 209, 266} \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))+\frac {1}{4} d^4 x^4 (a+b \arctan (c x))-\frac {11 b d^4 \arctan (c x)}{8 c^4}-\frac {1}{56} b c^3 d^4 x^7+\frac {11 b d^4 x}{8 c^3}+\frac {2}{21} i b c^2 d^4 x^6+\frac {24 i b d^4 x^2}{35 c^2}-\frac {24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac {9}{40} b c d^4 x^5-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4 \]

[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 12

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

Rule 45

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 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 649

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 1816

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 4992

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]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^4 x^4 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-(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 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\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 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\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 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\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 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\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 \arctan (c x)}{8 c^4}+\frac {1}{4} d^4 x^4 (a+b \arctan (c x))+\frac {4}{5} i c d^4 x^5 (a+b \arctan (c x))-c^2 d^4 x^6 (a+b \arctan (c x))-\frac {4}{7} i c^3 d^4 x^7 (a+b \arctan (c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \arctan (c x))-\frac {24 i b d^4 \log \left (1+c^2 x^2\right )}{35 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.22 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {11 b d^4 x}{8 c^3}+\frac {24 i b d^4 x^2}{35 c^2}-\frac {11 b d^4 x^3}{24 c}+\frac {1}{4} a d^4 x^4-\frac {12}{35} i b d^4 x^4+\frac {4}{5} i a c d^4 x^5+\frac {9}{40} b c d^4 x^5-a c^2 d^4 x^6+\frac {2}{21} i b c^2 d^4 x^6-\frac {4}{7} i a c^3 d^4 x^7-\frac {1}{56} b c^3 d^4 x^7+\frac {1}{8} a c^4 d^4 x^8-\frac {11 b d^4 \arctan (c x)}{8 c^4}+\frac {1}{4} b d^4 x^4 \arctan (c x)+\frac {4}{5} i b c d^4 x^5 \arctan (c x)-b c^2 d^4 x^6 \arctan (c x)-\frac {4}{7} i b c^3 d^4 x^7 \arctan (c x)+\frac {1}{8} b c^4 d^4 x^8 \arctan (c x)-\frac {24 i b d^4 \log \left (1+c^2 x^2\right )}{35 c^4} \]

[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

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.80

method	result	size
parts	\(d^{4} a \left (\frac {1}{8} c^{4} x^{8}-\frac {4}{7} i c^{3} x^{7}-c^{2} x^{6}+\frac {4}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{4} b \left (\frac {\arctan \left (c x \right ) c^{8} x^{8}}{8}-\frac {4 i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\arctan \left (c x \right ) c^{6} x^{6}+\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {11 c x}{8}-\frac {c^{7} x^{7}}{56}+\frac {2 i c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}-\frac {12 i c^{4} x^{4}}{35}-\frac {11 c^{3} x^{3}}{24}+\frac {24 i c^{2} x^{2}}{35}-\frac {24 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {11 \arctan \left (c x \right )}{8}\right )}{c^{4}}\)	\(190\)
derivativedivides	\(\frac {d^{4} a \left (\frac {1}{8} c^{8} x^{8}-\frac {4}{7} i c^{7} x^{7}-c^{6} x^{6}+\frac {4}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{4} b \left (\frac {\arctan \left (c x \right ) c^{8} x^{8}}{8}-\frac {4 i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\arctan \left (c x \right ) c^{6} x^{6}+\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {11 c x}{8}-\frac {c^{7} x^{7}}{56}+\frac {2 i c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}-\frac {12 i c^{4} x^{4}}{35}-\frac {11 c^{3} x^{3}}{24}+\frac {24 i c^{2} x^{2}}{35}-\frac {24 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {11 \arctan \left (c x \right )}{8}\right )}{c^{4}}\)	\(196\)
default	\(\frac {d^{4} a \left (\frac {1}{8} c^{8} x^{8}-\frac {4}{7} i c^{7} x^{7}-c^{6} x^{6}+\frac {4}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{4} b \left (\frac {\arctan \left (c x \right ) c^{8} x^{8}}{8}-\frac {4 i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\arctan \left (c x \right ) c^{6} x^{6}+\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {11 c x}{8}-\frac {c^{7} x^{7}}{56}+\frac {2 i c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}-\frac {12 i c^{4} x^{4}}{35}-\frac {11 c^{3} x^{3}}{24}+\frac {24 i c^{2} x^{2}}{35}-\frac {24 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {11 \arctan \left (c x \right )}{8}\right )}{c^{4}}\)	\(196\)
parallelrisch	\(-\frac {-105 b \,c^{8} d^{4} \arctan \left (c x \right ) x^{8}-80 i x^{6} b \,c^{6} d^{4}-105 a \,c^{8} d^{4} x^{8}+288 i x^{4} b \,c^{4} d^{4}+15 b \,c^{7} d^{4} x^{7}+480 i c^{7} b \,d^{4} \arctan \left (c x \right ) x^{7}+840 b \,c^{6} d^{4} \arctan \left (c x \right ) x^{6}+576 i b \,d^{4} \ln \left (c^{2} x^{2}+1\right )+840 a \,c^{6} d^{4} x^{6}-672 i c^{5} b \,d^{4} \arctan \left (c x \right ) x^{5}-189 b \,c^{5} d^{4} x^{5}-672 i x^{5} a \,c^{5} d^{4}-210 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{4}-210 a \,c^{4} d^{4} x^{4}+385 b \,c^{3} d^{4} x^{3}-576 i x^{2} b \,c^{2} d^{4}+480 i x^{7} a \,c^{7} d^{4}-1155 b c \,d^{4} x +1155 b \,d^{4} \arctan \left (c x \right )}{840 c^{4}}\)	\(261\)
risch	\(-\frac {i d^{4} c^{2} x^{6} b \ln \left (-i c x +1\right )}{2}+\frac {d^{4} c^{4} a \,x^{8}}{8}-\frac {4 i d^{4} c^{3} a \,x^{7}}{7}+\frac {2 d^{4} c^{3} b \,x^{7} \ln \left (-i c x +1\right )}{7}-\frac {b \,c^{3} d^{4} x^{7}}{56}-\frac {i d^{4} b \left (35 c^{4} x^{8}-160 i c^{3} x^{7}-280 c^{2} x^{6}+224 i c \,x^{5}+70 x^{4}\right ) \ln \left (i c x +1\right )}{560}-d^{4} c^{2} a \,x^{6}-\frac {24 i d^{4} b \ln \left (148225 c^{2} x^{2}+148225\right )}{35 c^{4}}-\frac {2 d^{4} c b \,x^{5} \ln \left (-i c x +1\right )}{5}+\frac {i d^{4} x^{4} b \ln \left (-i c x +1\right )}{8}+\frac {9 b c \,d^{4} x^{5}}{40}+\frac {i d^{4} c^{4} b \,x^{8} \ln \left (-i c x +1\right )}{16}+\frac {d^{4} a \,x^{4}}{4}+\frac {4 i d^{4} c a \,x^{5}}{5}+\frac {24 i b \,d^{4} x^{2}}{35 c^{2}}-\frac {11 b \,d^{4} x^{3}}{24 c}+\frac {2 i b \,c^{2} d^{4} x^{6}}{21}+\frac {11 b \,d^{4} x}{8 c^{3}}-\frac {11 b \,d^{4} \arctan \left (c x \right )}{8 c^{4}}-\frac {12 i b \,d^{4} x^{4}}{35}\)	\(324\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.97 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {210 \, a c^{8} d^{4} x^{8} - 30 \, {\left (32 i \, a + b\right )} c^{7} d^{4} x^{7} - 80 \, {\left (21 \, a - 2 i \, b\right )} c^{6} d^{4} x^{6} - 42 \, {\left (-32 i \, a - 9 \, b\right )} c^{5} d^{4} x^{5} + 12 \, {\left (35 \, a - 48 i \, b\right )} c^{4} d^{4} x^{4} - 770 \, b c^{3} d^{4} x^{3} + 1152 i \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 2307 i \, b d^{4} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b d^{4} \log \left (\frac {c x - i}{c}\right ) - 3 \, {\left (-35 i \, b c^{8} d^{4} x^{8} - 160 \, b c^{7} d^{4} x^{7} + 280 i \, b c^{6} d^{4} x^{6} + 224 \, b c^{5} d^{4} x^{5} - 70 i \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{1680 \, c^{4}} \]

[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 - 30*(32*I*a + b)*c^7*d^4*x^7 - 80*(21*a - 2*I*b)*c^6*d^4*x^6 - 42*(-32*I*a - 9*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 - 2307
*I*b*d^4*log((c*x + I)/c) + 3*I*b*d^4*log((c*x - I)/c) - 3*(-35*I*b*c^8*d^4*x^8 - 160*b*c^7*d^4*x^7 + 280*I*b*
c^6*d^4*x^6 + 224*b*c^5*d^4*x^5 - 70*I*b*c^4*d^4*x^4)*log(-(c*x + I)/(c*x - I)))/c^4

Sympy [A] (verification not implemented)

Time = 3.38 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.63 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {a c^{4} d^{4} x^{8}}{8} - \frac {11 b d^{4} x^{3}}{24 c} + \frac {24 i b d^{4} x^{2}}{35 c^{2}} + \frac {11 b d^{4} x}{8 c^{3}} + \frac {b d^{4} \left (\frac {i \log {\left (5893 b c d^{4} x - 5893 i b d^{4} \right )}}{560} - \frac {1471 i \log {\left (5893 b c d^{4} x + 5893 i b d^{4} \right )}}{1260}\right )}{c^{4}} + x^{7} \left (- \frac {4 i a c^{3} d^{4}}{7} - \frac {b c^{3} d^{4}}{56}\right ) + x^{6} \left (- a c^{2} d^{4} + \frac {2 i b c^{2} d^{4}}{21}\right ) + x^{5} \cdot \left (\frac {4 i a c d^{4}}{5} + \frac {9 b c d^{4}}{40}\right ) + x^{4} \left (\frac {a d^{4}}{4} - \frac {12 i b d^{4}}{35}\right ) + \left (- \frac {i b c^{4} d^{4} x^{8}}{16} - \frac {2 b c^{3} d^{4} x^{7}}{7} + \frac {i b c^{2} d^{4} x^{6}}{2} + \frac {2 b c d^{4} x^{5}}{5} - \frac {i b d^{4} x^{4}}{8}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (315 i b c^{8} d^{4} x^{8} + 1440 b c^{7} d^{4} x^{7} - 2520 i b c^{6} d^{4} x^{6} - 2016 b c^{5} d^{4} x^{5} + 630 i b c^{4} d^{4} x^{4} - 1037 i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{5040 c^{4}} \]

[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) + b*d**4*(I*log
(5893*b*c*d**4*x - 5893*I*b*d**4)/560 - 1471*I*log(5893*b*c*d**4*x + 5893*I*b*d**4)/1260)/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**4/4
0) + 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) + (315*I*b*c**8*d**4*x**8 + 1440*b*c**7*d**4*x**7 - 2520
*I*b*c**6*d**4*x**6 - 2016*b*c**5*d**4*x**5 + 630*I*b*c**4*d**4*x**4 - 1037*I*b*d**4)*log(-I*c*x + 1)/(5040*c*
*4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.42 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {1}{8} \, a c^{4} d^{4} x^{8} - \frac {4}{7} i \, a c^{3} d^{4} x^{7} - a c^{2} d^{4} x^{6} + \frac {4}{5} i \, a c d^{4} x^{5} + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} - \frac {1}{21} i \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac {1}{4} \, a d^{4} x^{4} - \frac {1}{15} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac {1}{5} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{4} \]

[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

Giac [F]

\[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )} x^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 2.78 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.91 \[ \int x^3 (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {c^4\,d^4\,\left (105\,a\,x^8+105\,b\,x^8\,\mathrm {atan}\left (c\,x\right )\right )}{840}+\frac {d^4\,\left (210\,a\,x^4+210\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,288{}\mathrm {i}\right )}{840}-\frac {\frac {d^4\,\left (1155\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,576{}\mathrm {i}\right )}{840}+\frac {11\,b\,c^3\,d^4\,x^3}{24}-\frac {11\,b\,c\,d^4\,x}{8}-\frac {b\,c^2\,d^4\,x^2\,24{}\mathrm {i}}{35}}{c^4}+\frac {c\,d^4\,\left (a\,x^5\,672{}\mathrm {i}+189\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,672{}\mathrm {i}\right )}{840}-\frac {c^3\,d^4\,\left (a\,x^7\,480{}\mathrm {i}+15\,b\,x^7+b\,x^7\,\mathrm {atan}\left (c\,x\right )\,480{}\mathrm {i}\right )}{840}-\frac {c^2\,d^4\,\left (840\,a\,x^6+840\,b\,x^6\,\mathrm {atan}\left (c\,x\right )-b\,x^6\,80{}\mathrm {i}\right )}{840} \]

[In]

int(x^3*(a + b*atan(c*x))*(d + c*d*x*1i)^4,x)

[Out]

(d^4*(210*a*x^4 - b*x^4*288i + 210*b*x^4*atan(c*x)))/840 - ((d^4*(1155*b*atan(c*x) + b*log(c^2*x^2 + 1)*576i))
/840 - (b*c^2*d^4*x^2*24i)/35 + (11*b*c^3*d^4*x^3)/24 - (11*b*c*d^4*x)/8)/c^4 + (c^4*d^4*(105*a*x^8 + 105*b*x^
8*atan(c*x)))/840 + (c*d^4*(a*x^5*672i + 189*b*x^5 + b*x^5*atan(c*x)*672i))/840 - (c^3*d^4*(a*x^7*480i + 15*b*
x^7 + b*x^7*atan(c*x)*480i))/840 - (c^2*d^4*(840*a*x^6 - b*x^6*80i + 840*b*x^6*atan(c*x)))/840