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

Optimal. Leaf size=203 \[ \frac{1}{2} i b d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^4 \text{PolyLog}(2,i c x)+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 x+a d^4 \log (x)-\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} i b c^2 d^4 x^2-\frac{8}{3} i b d^4 \log \left (c^2 x^2+1\right )+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x) \]

[Out]

(4*I)*a*c*d^4*x + (13*b*c*d^4*x)/4 + ((2*I)/3)*b*c^2*d^4*x^2 - (b*c^3*d^4*x^3)/12 - (13*b*d^4*ArcTan[c*x])/4 +
(4*I)*b*c*d^4*x*ArcTan[c*x] - 3*c^2*d^4*x^2*(a + b*ArcTan[c*x]) - ((4*I)/3)*c^3*d^4*x^3*(a + b*ArcTan[c*x]) +
(c^4*d^4*x^4*(a + b*ArcTan[c*x]))/4 + a*d^4*Log[x] - ((8*I)/3)*b*d^4*Log[1 + c^2*x^2] + (I/2)*b*d^4*PolyLog[2
, (-I)*c*x] - (I/2)*b*d^4*PolyLog[2, I*c*x]

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Rubi [A]  time = 0.210705, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {4876, 4846, 260, 4848, 2391, 4852, 321, 203, 266, 43, 302} \[ \frac{1}{2} i b d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^4 \text{PolyLog}(2,i c x)+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 x+a d^4 \log (x)-\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} i b c^2 d^4 x^2-\frac{8}{3} i b d^4 \log \left (c^2 x^2+1\right )+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*I)*a*c*d^4*x + (13*b*c*d^4*x)/4 + ((2*I)/3)*b*c^2*d^4*x^2 - (b*c^3*d^4*x^3)/12 - (13*b*d^4*ArcTan[c*x])/4 +
(4*I)*b*c*d^4*x*ArcTan[c*x] - 3*c^2*d^4*x^2*(a + b*ArcTan[c*x]) - ((4*I)/3)*c^3*d^4*x^3*(a + b*ArcTan[c*x]) +
(c^4*d^4*x^4*(a + b*ArcTan[c*x]))/4 + a*d^4*Log[x] - ((8*I)/3)*b*d^4*Log[1 + c^2*x^2] + (I/2)*b*d^4*PolyLog[2
, (-I)*c*x] - (I/2)*b*d^4*PolyLog[2, I*c*x]

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 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]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 321

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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int \frac{(d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=\int \left (4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-6 c^2 d^4 x \left (a+b \tan ^{-1}(c x)\right )-4 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx+\left (4 i c d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (6 c^2 d^4\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (4 i c^3 d^4\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=4 i a c d^4 x-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)+\frac{1}{2} \left (i b d^4\right ) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} \left (i b d^4\right ) \int \frac{\log (1+i c x)}{x} \, dx+\left (4 i b c d^4\right ) \int \tan ^{-1}(c x) \, dx+\left (3 b c^3 d^4\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{1}{3} \left (4 i b c^4 d^4\right ) \int \frac{x^3}{1+c^2 x^2} \, dx-\frac{1}{4} \left (b c^5 d^4\right ) \int \frac{x^4}{1+c^2 x^2} \, dx\\ &=4 i a c d^4 x+3 b c d^4 x+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)+\frac{1}{2} i b d^4 \text{Li}_2(-i c x)-\frac{1}{2} i b d^4 \text{Li}_2(i c x)-\left (3 b c d^4\right ) \int \frac{1}{1+c^2 x^2} \, dx-\left (4 i b c^2 d^4\right ) \int \frac{x}{1+c^2 x^2} \, dx+\frac{1}{3} \left (2 i b c^4 d^4\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{4} \left (b c^5 d^4\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=4 i a c d^4 x+\frac{13}{4} b c d^4 x-\frac{1}{12} b c^3 d^4 x^3-3 b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)-2 i b d^4 \log \left (1+c^2 x^2\right )+\frac{1}{2} i b d^4 \text{Li}_2(-i c x)-\frac{1}{2} i b d^4 \text{Li}_2(i c x)-\frac{1}{4} \left (b c d^4\right ) \int \frac{1}{1+c^2 x^2} \, dx+\frac{1}{3} \left (2 i b c^4 d^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=4 i a c d^4 x+\frac{13}{4} b c d^4 x+\frac{2}{3} i b c^2 d^4 x^2-\frac{1}{12} b c^3 d^4 x^3-\frac{13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)-\frac{8}{3} i b d^4 \log \left (1+c^2 x^2\right )+\frac{1}{2} i b d^4 \text{Li}_2(-i c x)-\frac{1}{2} i b d^4 \text{Li}_2(i c x)\\ \end{align*}

Mathematica [A]  time = 0.141025, size = 174, normalized size = 0.86 \[ \frac{1}{12} d^4 \left (6 i b \text{PolyLog}(2,-i c x)-6 i b \text{PolyLog}(2,i c x)+3 a c^4 x^4-16 i a c^3 x^3-36 a c^2 x^2+48 i a c x+12 a \log (x)-b c^3 x^3+8 i b c^2 x^2-32 i b \log \left (c^2 x^2+1\right )+3 b c^4 x^4 \tan ^{-1}(c x)-16 i b c^3 x^3 \tan ^{-1}(c x)-36 b c^2 x^2 \tan ^{-1}(c x)+39 b c x+48 i b c x \tan ^{-1}(c x)-39 b \tan ^{-1}(c x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^4*((48*I)*a*c*x + 39*b*c*x - 36*a*c^2*x^2 + (8*I)*b*c^2*x^2 - (16*I)*a*c^3*x^3 - b*c^3*x^3 + 3*a*c^4*x^4 -
39*b*ArcTan[c*x] + (48*I)*b*c*x*ArcTan[c*x] - 36*b*c^2*x^2*ArcTan[c*x] - (16*I)*b*c^3*x^3*ArcTan[c*x] + 3*b*c^
4*x^4*ArcTan[c*x] + 12*a*Log[x] - (32*I)*b*Log[1 + c^2*x^2] + (6*I)*b*PolyLog[2, (-I)*c*x] - (6*I)*b*PolyLog[2
, I*c*x]))/12

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Maple [A]  time = 0.041, size = 260, normalized size = 1.3 \begin{align*}{\frac{i}{2}}{d}^{4}b{\it dilog} \left ( 1+icx \right ) +{\frac{{d}^{4}a{c}^{4}{x}^{4}}{4}}+{\frac{i}{2}}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -3\,{d}^{4}a{c}^{2}{x}^{2}+{d}^{4}a\ln \left ( cx \right ) +{\frac{2\,i}{3}}b{c}^{2}{d}^{4}{x}^{2}+{\frac{{d}^{4}b\arctan \left ( cx \right ){c}^{4}{x}^{4}}{4}}-{\frac{4\,i}{3}}{d}^{4}a{c}^{3}{x}^{3}-3\,{d}^{4}b\arctan \left ( cx \right ){c}^{2}{x}^{2}+{d}^{4}b\arctan \left ( cx \right ) \ln \left ( cx \right ) -{\frac{8\,i}{3}}b{d}^{4}\ln \left ({c}^{2}{x}^{2}+1 \right ) +4\,iac{d}^{4}x-{\frac{i}{2}}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +4\,ibc{d}^{4}x\arctan \left ( cx \right ) +{\frac{13\,bc{d}^{4}x}{4}}-{\frac{b{c}^{3}{d}^{4}{x}^{3}}{12}}-{\frac{4\,i}{3}}{d}^{4}b\arctan \left ( cx \right ){c}^{3}{x}^{3}-{\frac{i}{2}}{d}^{4}b{\it dilog} \left ( 1-icx \right ) -{\frac{13\,b{d}^{4}\arctan \left ( cx \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*d^4*b*dilog(1+I*c*x)+1/4*d^4*a*c^4*x^4+1/2*I*d^4*b*ln(c*x)*ln(1+I*c*x)-3*d^4*a*c^2*x^2+d^4*a*ln(c*x)+2/3
*I*b*c^2*d^4*x^2+1/4*d^4*b*arctan(c*x)*c^4*x^4-4/3*I*d^4*a*c^3*x^3-3*d^4*b*arctan(c*x)*c^2*x^2+d^4*b*arctan(c*
x)*ln(c*x)-8/3*I*b*d^4*ln(c^2*x^2+1)+4*I*a*c*d^4*x-1/2*I*d^4*b*ln(c*x)*ln(1-I*c*x)+4*I*b*c*d^4*x*arctan(c*x)+1
3/4*b*c*d^4*x-1/12*b*c^3*d^4*x^3-4/3*I*d^4*b*arctan(c*x)*c^3*x^3-1/2*I*d^4*b*dilog(1-I*c*x)-13/4*b*d^4*arctan(
c*x)

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Maxima [A]  time = 2.20149, size = 308, normalized size = 1.52 \begin{align*} \frac{1}{4} \, a c^{4} d^{4} x^{4} - \frac{4}{3} i \, a c^{3} d^{4} x^{3} - \frac{1}{12} \, b c^{3} d^{4} x^{3} - 3 \, a c^{2} d^{4} x^{2} + \frac{2}{3} i \, b c^{2} d^{4} x^{2} + 4 i \, a c d^{4} x + \frac{13}{4} \, b c d^{4} x - \frac{1}{12} \,{\left (3 \, \pi + 8 i\right )} b d^{4} \log \left (c^{2} x^{2} + 1\right ) + b d^{4} \arctan \left (c x\right ) \log \left (x{\left | c \right |}\right ) + 2 i \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{4} - \frac{1}{2} i \, b d^{4}{\rm Li}_2\left (i \, c x + 1\right ) + \frac{1}{2} i \, b d^{4}{\rm Li}_2\left (-i \, c x + 1\right ) + a d^{4} \log \left (x\right ) + \frac{1}{12} \,{\left (3 \, b c^{4} d^{4} x^{4} - 16 i \, b c^{3} d^{4} x^{3} - 36 \, b c^{2} d^{4} x^{2} + 3 \, b d^{4}{\left (4 i \, \arctan \left (0, c\right ) - 13\right )}\right )} \arctan \left (c x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*c^4*d^4*x^4 - 4/3*I*a*c^3*d^4*x^3 - 1/12*b*c^3*d^4*x^3 - 3*a*c^2*d^4*x^2 + 2/3*I*b*c^2*d^4*x^2 + 4*I*a*c
*d^4*x + 13/4*b*c*d^4*x - 1/12*(3*pi + 8*I)*b*d^4*log(c^2*x^2 + 1) + b*d^4*arctan(c*x)*log(x*abs(c)) + 2*I*(2*
c*x*arctan(c*x) - log(c^2*x^2 + 1))*b*d^4 - 1/2*I*b*d^4*dilog(I*c*x + 1) + 1/2*I*b*d^4*dilog(-I*c*x + 1) + a*d
^4*log(x) + 1/12*(3*b*c^4*d^4*x^4 - 16*I*b*c^3*d^4*x^3 - 36*b*c^2*d^4*x^2 + 3*b*d^4*(4*I*arctan2(0, c) - 13))*
arctan(c*x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{2 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} +{\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{2 \, x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/2*(2*a*c^4*d^4*x^4 - 8*I*a*c^3*d^4*x^3 - 12*a*c^2*d^4*x^2 + 8*I*a*c*d^4*x + 2*a*d^4 + (I*b*c^4*d^4*
x^4 + 4*b*c^3*d^4*x^3 - 6*I*b*c^2*d^4*x^2 - 4*b*c*d^4*x + I*b*d^4)*log(-(c*x + I)/(c*x - I)))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{4} \left (\int \frac{a}{x}\, dx + \int 4 i a c\, dx + \int - 6 a c^{2} x\, dx + \int a c^{4} x^{3}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x}\, dx + \int - 4 i a c^{3} x^{2}\, dx + \int 4 i b c \operatorname{atan}{\left (c x \right )}\, dx + \int - 6 b c^{2} x \operatorname{atan}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \operatorname{atan}{\left (c x \right )}\, dx + \int - 4 i b c^{3} x^{2} \operatorname{atan}{\left (c x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**4*(Integral(a/x, x) + Integral(4*I*a*c, x) + Integral(-6*a*c**2*x, x) + Integral(a*c**4*x**3, x) + Integral
(b*atan(c*x)/x, x) + Integral(-4*I*a*c**3*x**2, x) + Integral(4*I*b*c*atan(c*x), x) + Integral(-6*b*c**2*x*ata
n(c*x), x) + Integral(b*c**4*x**3*atan(c*x), x) + Integral(-4*I*b*c**3*x**2*atan(c*x), x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, c d x + d\right )}^{4}{\left (b \arctan \left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((I*c*d*x + d)^4*(b*arctan(c*x) + a)/x, x)