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

Optimal. Leaf size=128 \[ \frac{3 b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}+\frac{3 i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 i b^3 \text{PolyLog}\left (4,-1+\frac{2}{1+i c x}\right )}{4 d}+\frac{\log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d} \]

[Out]

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

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Rubi [A]  time = 0.232061, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4868, 4884, 4994, 4998, 6610} \[ \frac{3 b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}+\frac{3 i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac{3 i b^3 \text{PolyLog}\left (4,-1+\frac{2}{1+i c x}\right )}{4 d}+\frac{\log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 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 4884

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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 4998

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a
+ b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 -
(2*I)/(I - c*x))^2, 0]

Rule 6610

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*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{x (d+i c d x)} \, dx &=\frac{\left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{(3 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{3 i b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{\left (3 i b^2 c\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{3 i b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{3 b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{\left (3 b^3 c\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{3 i b \left (a+b \tan ^{-1}(c x)\right )^2 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{3 b^2 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{3 i b^3 \text{Li}_4\left (-1+\frac{2}{1+i c x}\right )}{4 d}\\ \end{align*}

Mathematica [B]  time = 0.172057, size = 311, normalized size = 2.43 \[ \frac{6 b^2 \text{PolyLog}\left (3,\frac{c x+i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )+6 i b \text{PolyLog}\left (2,\frac{c x+i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-3 i b^3 \text{PolyLog}\left (4,\frac{c x+i}{-c x+i}\right )+12 a^2 b \log \left (\frac{2 i}{-c x+i}\right ) \tan ^{-1}(c x)+24 a^2 b \tan ^{-1}(c x) \tanh ^{-1}\left (\frac{c x+i}{c x-i}\right )+4 a^3 \log \left (\frac{2 i}{-c x+i}\right )+8 a^3 \tanh ^{-1}\left (\frac{c x+i}{c x-i}\right )+12 a b^2 \log \left (\frac{2 i}{-c x+i}\right ) \tan ^{-1}(c x)^2+24 a b^2 \tan ^{-1}(c x)^2 \tanh ^{-1}\left (\frac{c x+i}{c x-i}\right )+4 b^3 \log \left (\frac{2 i}{-c x+i}\right ) \tan ^{-1}(c x)^3+8 b^3 \tan ^{-1}(c x)^3 \tanh ^{-1}\left (\frac{c x+i}{c x-i}\right )}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*a^3*ArcTanh[(I + c*x)/(-I + c*x)] + 24*a^2*b*ArcTan[c*x]*ArcTanh[(I + c*x)/(-I + c*x)] + 24*a*b^2*ArcTan[c*
x]^2*ArcTanh[(I + c*x)/(-I + c*x)] + 8*b^3*ArcTan[c*x]^3*ArcTanh[(I + c*x)/(-I + c*x)] + 4*a^3*Log[(2*I)/(I -
c*x)] + 12*a^2*b*ArcTan[c*x]*Log[(2*I)/(I - c*x)] + 12*a*b^2*ArcTan[c*x]^2*Log[(2*I)/(I - c*x)] + 4*b^3*ArcTan
[c*x]^3*Log[(2*I)/(I - c*x)] + (6*I)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(I - c*x)] + 6*b^2*(a + b*Ar
cTan[c*x])*PolyLog[3, (I + c*x)/(I - c*x)] - (3*I)*b^3*PolyLog[4, (I + c*x)/(I - c*x)])/(4*d)

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Maple [C]  time = 0.442, size = 3393, normalized size = 26.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*I*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)
-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-3/2*I*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^
2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-3/2*I*a*b^2/d*
Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arcta
n(c*x)^2+3/2*I*a*b^2/d*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x
^2+1)+1))^2*arctan(c*x)^2-3/2*I*a*b^2/d*Pi*arctan(c*x)^2*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x
^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-3/2*I*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/
(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/2*I*b^3/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))
*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*
x)^3-1/2*I*b^3/d*Pi*arctan(c*x)^3*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*
c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))-3*a*b^2/d*arctan(c*x)^2*ln(c*x-I)+3*a*b^2/d*arctan(c*x)^2*ln(2
*I*(1+I*c*x)^2/(c^2*x^2+1))+3*a*b^2/d*arctan(c*x)^2*ln(c*x)+3*a*b^2/d*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)
^(1/2))-3*a*b^2/d*arctan(c*x)^2*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+3*a*b^2/d*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+
1)^(1/2))-2*I*a*b^2/d*arctan(c*x)^3-3*I*b^3/d*arctan(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*I*b^3/d*
arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/2*I*b^3/d*Pi*arctan(c*x)^3+3*b*a^2/d*arctan(c*x)*ln(c*x
)-1/2*I*b^3/d*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^3+1/2*I*b^3/d*Pi*csgn
(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^3+1/2*I*b^3/d*Pi*csgn(((1+I*c*x)^2/(
c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^3-I*b^3/d*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^
2/(c^2*x^2+1)+1))^2*arctan(c*x)^3-3*I*a*b^2/d*Pi*arctan(c*x)^2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*
x^2+1)+1))^2+3/2*I*a*b^2/d*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+3/
2*I*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-1/2*I*b^3/d*Pi*
csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c
*x)^3-1/2*I*b^3/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*
x^2+1)+1))^2*arctan(c*x)^3-1/2*I*b^3/d*Pi*arctan(c*x)^3*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^
2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2+1/2*I*b^3/d*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)
+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^3-1/2*I*b^3/d*Pi*csgn(I*((1+I*c
*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1)
)^2*arctan(c*x)^3+1/2*I*b^3/d*Pi*arctan(c*x)^3*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1
)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-3/2*I*a*b^2/d*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*
arctan(c*x)^2-3/2*I*a*b^2/d*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-1
/2*a^3/d*ln(c^2*x^2+1)+a^3/d*ln(c*x)-3/2*I*b*a^2/d*ln(c*x)*ln(1-I*c*x)+3/2*I*b*a^2/d*ln(c*x-I)*ln(-1/2*I*(c*x+
I))+9/2*I*a*b^2/d*Pi*arctan(c*x)^2-6*I*a*b^2/d*arctan(c*x)*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-6*I*a*b^2/d
*arctan(c*x)*polylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*b^3/d*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)
^2/(c^2*x^2+1)+1))^2*arctan(c*x)^3+3/2*I*b*a^2/d*ln(c*x)*ln(1+I*c*x)+3/2*I*a*b^2/d*Pi*csgn(I*((1+I*c*x)^2/(c^2
*x^2+1)-1))*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1)
)*arctan(c*x)^2-b^3/d*arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+b^3/d*arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+
1)^(1/2))+b^3/d*arctan(c*x)^3*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*b^3/d*arctan(c*x)*polylog(3,(1+I*c*x)/(c^2*x
^2+1)^(1/2))+6*b^3/d*arctan(c*x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-I*a^3/d*arctan(c*x)+6*a*b^2/d*polylog
(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*a*b^2/d*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+b^3/d*arctan(c*x)^3*ln(c*x)+
6*I*b^3/d*polylog(4,-(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2*I*b^3/d*arctan(c*x)^4+6*I*b^3/d*polylog(4,(1+I*c*x)/(c^2
*x^2+1)^(1/2))-3/2*I*a*b^2/d*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*
x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-b^3/d*arctan(c*x)^3*ln(c*x-I)+b^3/d*arctan(c*x)^3*
ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-3*b*a^2/d*arctan(c*x)*ln(c*x-I)+3/2*I*b*a^2/d*dilog(1+I*c*x)-3/2*I*b*a^2/d*dil
og(1-I*c*x)+3/2*I*b*a^2/d*dilog(-1/2*I*(c*x+I))-3/4*I*b*a^2/d*ln(c*x-I)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a^{3}{\left (\frac{\log \left (i \, c x + 1\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + \frac{-64 i \, b^{3} \arctan \left (c x\right )^{4} + 64 \, b^{3} \arctan \left (c x\right )^{3} \log \left (c^{2} x^{2} + 1\right ) + 4 i \, b^{3} \log \left (c^{2} x^{2} + 1\right )^{4} - i \,{\left (\frac{64 \, b^{3} \arctan \left (c x\right )^{4}}{d} + 96 \, b^{3} c^{2} \int \frac{x^{2} \arctan \left (c x\right )^{2} \log \left (c^{2} x^{2} + 1\right )}{c^{2} d x^{3} + d x}\,{d x} + \frac{3 \, b^{3} \log \left (c^{2} x^{2} + 1\right )^{4}}{d} + \frac{512 \, a b^{2} \arctan \left (c x\right )^{3}}{d} +{\left (\frac{\log \left (c^{2} x^{2} + 1\right )^{4}}{d} - \frac{4 \,{\left (\log \left (c^{2} x^{2} + 1\right )^{3} \log \left (-c^{2} x^{2}\right ) + 3 \,{\rm Li}_2\left (c^{2} x^{2} + 1\right ) \log \left (c^{2} x^{2} + 1\right )^{2} - 6 \, \log \left (c^{2} x^{2} + 1\right ){\rm Li}_{3}(c^{2} x^{2} + 1) + 6 \,{\rm Li}_{4}(c^{2} x^{2} + 1)\right )}}{d}\right )} b^{3} + \frac{768 \, a^{2} b \arctan \left (c x\right )^{2}}{d} + 96 \, b^{3} \int \frac{\arctan \left (c x\right )^{2} \log \left (c^{2} x^{2} + 1\right )}{c^{2} d x^{3} + d x}\,{d x}\right )} d + 16 \, d \int -\frac{12 \, b^{3} c x \arctan \left (c x\right )^{2} \log \left (c^{2} x^{2} + 1\right ) - 96 \, a b^{2} \arctan \left (c x\right )^{2} - 96 \, a^{2} b \arctan \left (c x\right ) + 4 \,{\left (3 \, b^{3} c^{2} x^{2} - 7 \, b^{3}\right )} \arctan \left (c x\right )^{3} - 3 \,{\left (b^{3} c^{2} x^{2} + b^{3}\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2}}{c^{2} d x^{3} + d x}\,{d x}}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*(log(I*c*x + 1)/d - log(x)/d) + 1/512*(-64*I*b^3*arctan(c*x)^4 + 64*b^3*arctan(c*x)^3*log(c^2*x^2 + 1) +
16*b^3*arctan(c*x)*log(c^2*x^2 + 1)^3 + 4*I*b^3*log(c^2*x^2 + 1)^4 - I*(64*b^3*arctan(c*x)^4/d + 6144*b^3*c^2*
integrate(1/64*x^2*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^3 + d*x), x) + 3*b^3*log(c^2*x^2 + 1)^4/d + 512*a*b
^2*arctan(c*x)^3/d + 768*a^2*b*arctan(c*x)^2/d + 6144*b^3*integrate(1/64*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d
*x^3 + d*x), x) - 512*b^3*integrate(1/64*log(c^2*x^2 + 1)^3/(c^2*d*x^3 + d*x), x))*d - 512*d*integrate(1/32*(1
2*b^3*c*x*arctan(c*x)^2*log(c^2*x^2 + 1) + b^3*c*x*log(c^2*x^2 + 1)^3 - 96*a*b^2*arctan(c*x)^2 - 96*a^2*b*arct
an(c*x) + 4*(3*b^3*c^2*x^2 - 7*b^3)*arctan(c*x)^3 + 3*(b^3*c^2*x^2 - b^3)*arctan(c*x)*log(c^2*x^2 + 1)^2)/(c^2
*d*x^3 + d*x), x))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (i \, c d x + d\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c*x) + a)^3/((I*c*d*x + d)*x), x)