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

Optimal. Leaf size=147 \[ \frac{3 i b^3 \text{PolyLog}\left (2,1-\frac{2}{1+\frac{i c}{x}}\right )}{2 c^2}+\frac{3 b^2 \log \left (\frac{2}{1+\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )}{c^2}+\frac{3 i b \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c^2}-\frac{\left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3}{2 c^2}-\frac{\left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3}{2 x^2}+\frac{3 b \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c x} \]

[Out]

(((3*I)/2)*b*(a + b*ArcCot[x/c])^2)/c^2 + (3*b*(a + b*ArcCot[x/c])^2)/(2*c*x) - (a + b*ArcCot[x/c])^3/(2*c^2)
- (a + b*ArcCot[x/c])^3/(2*x^2) + (3*b^2*(a + b*ArcCot[x/c])*Log[2/(1 + (I*c)/x)])/c^2 + (((3*I)/2)*b^3*PolyLo
g[2, 1 - 2/(1 + (I*c)/x)])/c^2

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Rubi [F]  time = 2.25451, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTan[c/x])^3/x^3,x]

[Out]

(((3*I)/64)*b^3*(1 - (I*c)/x)^2)/c^2 - (3*a*b^2*(1 + (I*c)/x)^2)/(16*c^2) - (((3*I)/64)*b^3*(1 + (I*c)/x)^2)/c
^2 - (((3*I)/8)*a^2*b)/x^2 - (3*a*b^2)/(8*x^2) + (3*a^2*b)/(4*c*x) - (3*b^3)/(2*c*x) + (((3*I)/4)*a^2*b*Log[I
- c/x])/c^2 + (3*a*b^2*Log[I - c/x])/(8*c^2) - (3*a*b^2*(1 - (I*c)/x)*Log[1 - (I*c)/x])/(4*c^2) + (((3*I)/4)*b
^3*(1 - (I*c)/x)*Log[1 - (I*c)/x])/c^2 + (3*a*b^2*Log[1 - (I*c)/x])/(8*x^2) - (3*b^2*(1 - (I*c)/x)^2*(2*a + I*
b*Log[1 - (I*c)/x]))/(32*c^2) + (((3*I)/8)*b*(1 - (I*c)/x)*(2*a + I*b*Log[1 - (I*c)/x])^2)/c^2 - (((3*I)/32)*b
*(1 - (I*c)/x)^2*(2*a + I*b*Log[1 - (I*c)/x])^2)/c^2 - ((1 - (I*c)/x)*(2*a + I*b*Log[1 - (I*c)/x])^3)/(8*c^2)
+ ((1 - (I*c)/x)^2*(2*a + I*b*Log[1 - (I*c)/x])^3)/(16*c^2) - (9*a*b^2*(1 + (I*c)/x)*Log[1 + (I*c)/x])/(4*c^2)
- (((3*I)/4)*b^3*(1 + (I*c)/x)*Log[1 + (I*c)/x])/c^2 + (3*a*b^2*(1 + (I*c)/x)^2*Log[1 + (I*c)/x])/(8*c^2) + (
((3*I)/32)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x])/c^2 + (((3*I)/4)*a^2*b*Log[1 + (I*c)/x])/x^2 + (3*a*b^2*Log[1
+ (I*c)/x])/(8*x^2) - (3*a*b^2*Log[1 - (I*c)/x]*Log[1 + (I*c)/x])/(4*x^2) + (3*a*b^2*(1 + (I*c)/x)*Log[1 + (I
*c)/x]^2)/(4*c^2) + (((3*I)/8)*b^3*(1 + (I*c)/x)*Log[1 + (I*c)/x]^2)/c^2 - (3*a*b^2*(1 + (I*c)/x)^2*Log[1 + (I
*c)/x]^2)/(8*c^2) - (((3*I)/32)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x]^2)/c^2 - ((I/8)*b^3*(1 + (I*c)/x)*Log[1 +
(I*c)/x]^3)/c^2 + ((I/16)*b^3*(1 + (I*c)/x)^2*Log[1 + (I*c)/x]^3)/c^2 + (3*a*b^2*Log[I + c/x])/(8*c^2) - (3*a
*b^2*Log[1 - (I*c)/x]*Log[c - I*x])/(4*c^2) - (3*a*b^2*Log[1 + (I*c)/x]*Log[c + I*x])/(4*c^2) + (3*a*b^2*Log[(
c - I*x)/(2*c)]*Log[c + I*x])/(4*c^2) + (3*a*b^2*Log[c - I*x]*Log[(c + I*x)/(2*c)])/(4*c^2) - (3*a*b^2*Log[c +
I*x]*Log[((-I)*x)/c])/(4*c^2) - (3*a*b^2*Log[c - I*x]*Log[(I*x)/c])/(4*c^2) + (3*a*b^2*PolyLog[2, (c - I*x)/(
2*c)])/(4*c^2) + (3*a*b^2*PolyLog[2, (c + I*x)/(2*c)])/(4*c^2) + (3*a*b^2*PolyLog[2, ((-I)*c)/x])/(4*c^2) + (3
*a*b^2*PolyLog[2, (I*c)/x])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 - (I*x)/c])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 + (I*x
)/c])/(4*c^2) + ((3*I)/8)*b^3*Defer[Int][(Log[1 - (I*c)/x]^2*Log[1 + (I*c)/x])/x^3, x] - ((3*I)/8)*b^3*Defer[I
nt][(Log[1 - (I*c)/x]*Log[1 + (I*c)/x]^2)/x^3, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3}{x^3} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 x^3}+\frac{3 i b \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 x^3}-\frac{3 i b^2 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 x^3}+\frac{i b^3 \log ^3\left (1+\frac{i c}{x}\right )}{8 x^3}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{x^3} \, dx+\frac{1}{8} (3 i b) \int \frac{\left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^2\right ) \int \frac{\left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{1}{8} \left (i b^3\right ) \int \frac{\log ^3\left (1+\frac{i c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int x (2 a+i b \log (1-i c x))^3 \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} (3 i b) \int \left (-\frac{4 a^2 \log \left (1+\frac{i c}{x}\right )}{x^3}-\frac{4 i a b \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3}+\frac{b^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3}\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int \left (-\frac{2 i a \log ^2\left (1+\frac{i c}{x}\right )}{x^3}+\frac{b \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3}\right ) \, dx-\frac{1}{8} \left (i b^3\right ) \operatorname{Subst}\left (\int x \log ^3(1+i c x) \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \left (-\frac{i (2 a+i b \log (1-i c x))^3}{c}+\frac{i (1-i c x) (2 a+i b \log (1-i c x))^3}{c}\right ) \, dx,x,\frac{1}{x}\right )\right )-\frac{1}{2} \left (3 i a^2 b\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{1}{2} \left (3 a b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{i \log ^3(1+i c x)}{c}-\frac{i (1+i c x) \log ^3(1+i c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx\\ &=-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{1}{2} \left (3 i a^2 b\right ) \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c \log \left (1-\frac{i c}{x}\right )}{2 (c-i x) x^3} \, dx-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c \log \left (1+\frac{i c}{x}\right )}{2 (c+i x) x^3} \, dx+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^3 \, dx,x,\frac{1}{x}\right )}{8 c}-\frac{i \operatorname{Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^3 \, dx,x,\frac{1}{x}\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log ^3(1+i c x) \, dx,x,\frac{1}{x}\right )}{8 c}-\frac{b^3 \operatorname{Subst}\left (\int (1+i c x) \log ^3(1+i c x) \, dx,x,\frac{1}{x}\right )}{8 c}\\ &=\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (\frac{i \log ^2(1+i c x)}{c}-\frac{i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{\operatorname{Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-\frac{i c}{x}\right )}{8 c^2}+\frac{\operatorname{Subst}\left (\int x (2 a+i b \log (x))^3 \, dx,x,1-\frac{i c}{x}\right )}{8 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+\frac{i c}{x}\right )}{8 c^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+\frac{i c}{x}\right )}{8 c^2}+\frac{1}{4} \left (3 a^2 b c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 a b^2 c\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{(c-i x) x^3} \, dx-\frac{1}{4} \left (3 a b^2 c\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{(c+i x) x^3} \, dx\\ &=-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{(3 i b) \operatorname{Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-\frac{i c}{x}\right )}{16 c^2}+\frac{(3 i b) \operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-\frac{i c}{x}\right )}{8 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{16 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{8 c^2}+\frac{\left (3 i a b^2\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (3 i a b^2\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}+\frac{1}{4} \left (3 a^2 b c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 a b^2 c\right ) \int \left (-\frac{i \log \left (1-\frac{i c}{x}\right )}{c^3 (c-i x)}+\frac{\log \left (1-\frac{i c}{x}\right )}{c x^3}+\frac{i \log \left (1-\frac{i c}{x}\right )}{c^2 x^2}-\frac{\log \left (1-\frac{i c}{x}\right )}{c^3 x}\right ) \, dx-\frac{1}{4} \left (3 a b^2 c\right ) \int \left (\frac{i \log \left (1+\frac{i c}{x}\right )}{c^3 (c+i x)}+\frac{\log \left (1+\frac{i c}{x}\right )}{c x^3}-\frac{i \log \left (1+\frac{i c}{x}\right )}{c^2 x^2}-\frac{\log \left (1+\frac{i c}{x}\right )}{c^3 x}\right ) \, dx\\ &=-\frac{3 i a^2 b}{8 x^2}+\frac{3 a^2 b}{4 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}-\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-\frac{i c}{x}\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}+\frac{\left (3 i a b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx}{4 c^2}-\frac{\left (3 i a b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{i c}{x}\right )}{16 c^2}-\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}-\frac{\left (3 i a b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{x^2} \, dx}{4 c}+\frac{\left (3 i a b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^2} \, dx}{4 c}\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 i a b^2}{2 c x}-\frac{3 b^3}{4 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (1-i c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{i c}{x}\right )}{2 c^2}+\frac{\left (3 i b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}+\frac{\left (3 i a b^2\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx}{4 c}-\frac{\left (3 i a b^2\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx}{4 c}+\frac{\left (3 i a b^2\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (3 i a b^2\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{2 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}+\frac{\left (3 i a b^2\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx}{4 c}-\frac{\left (3 i a b^2\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx}{4 c}+\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{9 a b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{\left (3 i a b^2\right ) \int \frac{\log (c-i x)}{c+i x} \, dx}{4 c^2}-\frac{\left (3 i a b^2\right ) \int \frac{\log (c+i x)}{c-i x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log (c-i x)}{x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log (c+i x)}{x} \, dx}{4 c^2}-\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}-\frac{3 a b^2}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{9 a b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}+\frac{3 a b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{3 a b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{\left (3 i a b^2\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx}{4 c^2}+\frac{\left (3 i a b^2\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx}{4 c^2}+\frac{\left (3 i a b^2\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx}{4 c^2}-\frac{\left (3 i a b^2\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx}{4 c^2}\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}-\frac{3 a b^2}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{9 a b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}+\frac{3 a b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{3 a b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1-\frac{i x}{c}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1+\frac{i x}{c}\right )}{4 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )}{4 c^2}\\ &=\frac{3 i b^3 \left (1-\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2}{64 c^2}-\frac{3 i a^2 b}{8 x^2}-\frac{3 a b^2}{8 x^2}+\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 i a^2 b \log \left (i-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{32 c^2}+\frac{3 i b \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{32 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{8 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3}{16 c^2}-\frac{9 a b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{32 c^2}+\frac{3 i a^2 b \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{3 i b^3 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{3 i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{32 c^2}-\frac{i b^3 \left (1+\frac{i c}{x}\right ) \log ^3\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i b^3 \left (1+\frac{i c}{x}\right )^2 \log ^3\left (1+\frac{i c}{x}\right )}{16 c^2}+\frac{3 a b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{3 a b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{3 a b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{3 a b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c-i x}{2 c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c+i x}{2 c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1-\frac{i x}{c}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1+\frac{i x}{c}\right )}{4 c^2}+\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 i b^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.282954, size = 178, normalized size = 1.21 \[ \frac{-3 i b^3 x^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (a c (3 b x-a c)+6 b^2 x^2 \log \left (\frac{1}{\sqrt{\frac{c^2}{x^2}+1}}\right )\right )-3 b \tan ^{-1}\left (\frac{c}{x}\right ) \left (a \left (a \left (c^2+x^2\right )-2 b c x\right )-2 b^2 x^2 \log \left (1+e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )\right )+3 b^2 (c-i x) \tan ^{-1}\left (\frac{c}{x}\right )^2 (b x-a (c+i x))+b^3 \left (-\left (c^2+x^2\right )\right ) \tan ^{-1}\left (\frac{c}{x}\right )^3}{2 c^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(3*b^2*(c - I*x)*(-(a*(c + I*x)) + b*x)*ArcTan[c/x]^2 - b^3*(c^2 + x^2)*ArcTan[c/x]^3 - 3*b*ArcTan[c/x]*(a*(-2
*b*c*x + a*(c^2 + x^2)) - 2*b^2*x^2*Log[1 + E^((2*I)*ArcTan[c/x])]) + a*(a*c*(-(a*c) + 3*b*x) + 6*b^2*x^2*Log[
1/Sqrt[1 + c^2/x^2]]) - (3*I)*b^3*x^2*PolyLog[2, -E^((2*I)*ArcTan[c/x])])/(2*c^2*x^2)

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Maple [B]  time = 0.095, size = 396, normalized size = 2.7 \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}-{\frac{{b}^{3}}{2\,{x}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{3}}-{\frac{{b}^{3}}{2\,{c}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{3}}+{\frac{3\,{b}^{3}}{2\,cx} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{3\,{b}^{3}}{2\,{c}^{2}}\arctan \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }-{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}\ln \left ({\frac{c}{x}}-i \right ) \ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }+{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}\ln \left ( -{\frac{i}{2}} \left ({\frac{c}{x}}+i \right ) \right ) \ln \left ({\frac{c}{x}}-i \right ) }-{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}\ln \left ({\frac{c}{x}}+i \right ) \ln \left ({\frac{i}{2}} \left ({\frac{c}{x}}-i \right ) \right ) }-{\frac{{\frac{3\,i}{8}}{b}^{3}}{{c}^{2}} \left ( \ln \left ({\frac{c}{x}}+i \right ) \right ) ^{2}}+{\frac{{\frac{3\,i}{8}}{b}^{3}}{{c}^{2}} \left ( \ln \left ({\frac{c}{x}}-i \right ) \right ) ^{2}}+{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}{\it dilog} \left ( -{\frac{i}{2}} \left ({\frac{c}{x}}+i \right ) \right ) }+{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}\ln \left ({\frac{c}{x}}+i \right ) \ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }-{\frac{{\frac{3\,i}{4}}{b}^{3}}{{c}^{2}}{\it dilog} \left ({\frac{i}{2}} \left ({\frac{c}{x}}-i \right ) \right ) }-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{3\,{a}^{2}b}{2\,cx}}+{\frac{3\,{a}^{2}b}{2\,{c}^{2}}\arctan \left ({\frac{x}{c}} \right ) }-{\frac{3\,a{b}^{2}}{2\,{x}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{3\,a{b}^{2}}{2\,{c}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}+3\,{\frac{a{b}^{2}}{cx}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{3\,a{b}^{2}}{2\,{c}^{2}}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*arctan(c/x)^3 + 3*a*b^2*arctan(c/x)^2 + 3*a^2*b*arctan(c/x) + a^3)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atan(c/x))**3/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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