3.148 \(\int x^2 (a+b \tan ^{-1}(\frac{c}{x}))^3 \, dx\)

Optimal. Leaf size=229 \[ -i b^2 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{2} b^3 c^3 \text{PolyLog}\left (3,-1+\frac{2}{1-\frac{i c}{x}}\right )+b^2 c^2 x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )-\frac{1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{2} b c^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+b c^3 \log \left (2-\frac{2}{1-\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} b c x^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{2} b^3 c^3 \log \left (\frac{c^2}{x^2}+1\right )+b^3 c^3 \log (x) \]

[Out]

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

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Rubi [F]  time = 3.52024, 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 = {} \[ \int x^2 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*ArcTan[c/x])^3,x]

[Out]

(-I/2)*a^2*b*c^2*x + (3*a*b^2*c^2*x)/4 + (a^2*b*c*x^2)/4 - ((3*I)/4)*a*b^2*c^3*Log[I - c/x] + (a*b^2*c^2*x*Log
[1 - (I*c)/x])/2 + (I/4)*a*b^2*c*x^2*Log[1 - (I*c)/x] + (b^2*c^2*(1 - (I*c)/x)*x*(2*a + I*b*Log[1 - (I*c)/x]))
/8 + (b*c^3*(2*a + I*b*Log[1 - (I*c)/x])^2)/16 + (I/8)*b*c^2*(1 - (I*c)/x)*x*(2*a + I*b*Log[1 - (I*c)/x])^2 +
(b*c*x^2*(2*a + I*b*Log[1 - (I*c)/x])^2)/16 + (I/24)*c^3*(2*a + I*b*Log[1 - (I*c)/x])^3 + (x^3*(2*a + I*b*Log[
1 - (I*c)/x])^3)/24 - (I/8)*b^3*c^2*(1 + (I*c)/x)*x*Log[1 + (I*c)/x] - (I/2)*a*b^2*c*x^2*Log[1 + (I*c)/x] - (I
/2)*a^2*b*x^3*Log[1 + (I*c)/x] + (a*b^2*x^3*Log[1 - (I*c)/x]*Log[1 + (I*c)/x])/2 + (I/4)*a*b^2*c^3*Log[1 + (I*
c)/x]^2 - (b^3*c^3*Log[1 + (I*c)/x]^2)/16 + (I/8)*b^3*c^2*(1 + (I*c)/x)*x*Log[1 + (I*c)/x]^2 - (b^3*c*x^2*Log[
1 + (I*c)/x]^2)/16 - (a*b^2*x^3*Log[1 + (I*c)/x]^2)/4 + (b^3*c^3*Log[1 + (I*c)/x]^3)/24 + (I/24)*b^3*x^3*Log[1
+ (I*c)/x]^3 - (a^2*b*c^3*Log[c - I*x])/2 + (I/4)*a*b^2*c^3*Log[c - I*x] - (I/2)*a*b^2*c^3*Log[1 - (I*c)/x]*L
og[c - I*x] - (I/4)*a*b^2*c^3*Log[c + I*x] + (I/2)*a*b^2*c^3*Log[1 + (I*c)/x]*Log[c + I*x] - (I/2)*a*b^2*c^3*L
og[(c - I*x)/(2*c)]*Log[c + I*x] + (I/2)*a*b^2*c^3*Log[c - I*x]*Log[(c + I*x)/(2*c)] - (b^3*c^3*Log[1 + (I*c)/
x]^2*Log[((-I)*c)/x])/8 + (b*c^3*(2*a + I*b*Log[1 - (I*c)/x])^2*Log[(I*c)/x])/8 + (b^3*c^3*Log[x])/4 + (I/2)*a
*b^2*c^3*Log[c + I*x]*Log[((-I)*x)/c] - (I/2)*a*b^2*c^3*Log[c - I*x]*Log[(I*x)/c] + (I/4)*b^2*c^3*(2*a + I*b*L
og[1 - (I*c)/x])*PolyLog[2, 1 - (I*c)/x] - (b^3*c^3*Log[1 + (I*c)/x]*PolyLog[2, 1 + (I*c)/x])/4 + (I/2)*a*b^2*
c^3*PolyLog[2, (c - I*x)/(2*c)] - (I/2)*a*b^2*c^3*PolyLog[2, (c + I*x)/(2*c)] + (I/2)*a*b^2*c^3*PolyLog[2, ((-
I)*c)/x] - (3*b^3*c^3*PolyLog[2, ((-I)*c)/x])/8 - (3*b^3*c^3*PolyLog[2, (I*c)/x])/8 - (I/2)*a*b^2*c^3*PolyLog[
2, 1 - (I*x)/c] + (I/2)*a*b^2*c^3*PolyLog[2, 1 + (I*x)/c] + (b^3*c^3*PolyLog[3, 1 - (I*c)/x])/4 + (b^3*c^3*Pol
yLog[3, 1 + (I*c)/x])/4 + ((3*I)/8)*b^3*Defer[Int][x^2*Log[1 - (I*c)/x]^2*Log[1 + (I*c)/x], x] - ((3*I)/8)*b^3
*Defer[Int][x^2*Log[1 - (I*c)/x]*Log[1 + (I*c)/x]^2, x]

Rubi steps

\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{3}{8} i b x^2 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i b^2 x^2 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 x^2 \log ^3\left (1+\frac{i c}{x}\right )\right ) \, dx\\ &=\frac{1}{8} \int x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3 \, dx+\frac{1}{8} (3 i b) \int x^2 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int x^2 \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (i b^3\right ) \int x^2 \log ^3\left (1+\frac{i c}{x}\right ) \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^3}{x^4} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} (3 i b) \int \left (-4 a^2 x^2 \log \left (1+\frac{i c}{x}\right )-4 i a b x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+b^2 x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int \left (-2 i a x^2 \log ^2\left (1+\frac{i c}{x}\right )+b x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )\right ) \, dx-\frac{1}{8} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^3(1+i c x)}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} \left (3 i a^2 b\right ) \int x^2 \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{4} \left (3 a b^2\right ) \int x^2 \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (3 a b^2\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^3 (1-i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^3 (1+i c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^4} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^2 \log \left (1-\frac{i c}{x}\right )}{3 (-c+i x)} \, dx-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^2 \log \left (1+\frac{i c}{x}\right )}{3 (-c-i x)} \, dx-\frac{1}{8} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^3} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (a^2 b c\right ) \int \frac{x}{1+\frac{i c}{x}} \, dx\\ &=\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^3} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} \left (a^2 b c\right ) \int \frac{x^2}{i c+x} \, dx+\frac{1}{2} \left (i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3 (1+i c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (a b^2 c\right ) \int \frac{x^2 \log \left (1-\frac{i c}{x}\right )}{-c+i x} \, dx-\frac{1}{2} \left (a b^2 c\right ) \int \frac{x^2 \log \left (1+\frac{i c}{x}\right )}{-c-i x} \, dx-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+\frac{i c}{x}\right )\\ &=\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} \left (a^2 b c\right ) \int \left (-i c+\frac{i c^2}{c-i x}+x\right ) \, dx-\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} \left (i a b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x^3}-\frac{i c \log (1+i c x)}{x^2}-\frac{c^2 \log (1+i c x)}{x}+\frac{c^3 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (a b^2 c\right ) \int \left (-c \log \left (1-\frac{i c}{x}\right )+\frac{c^2 \log \left (1-\frac{i c}{x}\right )}{c-i x}-i x \log \left (1-\frac{i c}{x}\right )\right ) \, dx-\frac{1}{2} \left (a b^2 c\right ) \int \left (-c \log \left (1+\frac{i c}{x}\right )+\frac{c^2 \log \left (1+\frac{i c}{x}\right )}{c+i x}+i x \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )} \, dx,x,1+\frac{i c}{x}\right )\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} \left (i a b^2 c\right ) \int x \log \left (1-\frac{i c}{x}\right ) \, dx-\frac{1}{2} \left (i a b^2 c\right ) \int x \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} \left (a b^2 c^2\right ) \int \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^2\right ) \int \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{4} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{8} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx+\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{2} \left (i a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \log (x)+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{1}{1-\frac{i c}{x}} \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{1}{1+\frac{i c}{x}} \, dx-\frac{1}{4} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{4} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (i c^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{4} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) (2 a+i b \log (x))}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{1}{\left (1-\frac{i c}{x}\right ) x} \, dx+\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{1}{\left (1+\frac{i c}{x}\right ) x} \, dx+\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log \left (1+\frac{i c}{x}\right )\right )-\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) \log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{2} \left (a b^2 c^4\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx-\frac{1}{2} \left (a b^2 c^4\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{3}{4} i a b^2 c^3 \log (x)+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{4} i b^2 c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) \text{Li}_2\left (1-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{x}{-i c+x} \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{x}{i c+x} \, dx-\frac{1}{4} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{i c}{x}+\frac{i c^2}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{1}{-i c+x} \, dx+\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{1}{i c+x} \, dx+\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{2} \left (a b^2 c^4\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx-\frac{1}{2} \left (a b^2 c^4\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{1}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2-\frac{3}{4} i a b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)+\frac{1}{2} i a b^2 c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{4} i b^2 c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) \text{Li}_2\left (1-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac{c}{c-i x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac{c}{c+i x}\right ) \, dx-\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{\log (c-i x)}{x} \, dx+\frac{1}{2} \left (i a b^2 c^3\right ) \int \frac{\log (c+i x)}{x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c-i x)}{c+i x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c+i x)}{c-i x} \, dx\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2-\frac{3}{4} i a b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)+\frac{1}{4} i a b^2 c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{4} i a b^2 c^3 \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{2} i a b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} i a b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{4} i b^2 c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) \text{Li}_2\left (1-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2-\frac{3}{4} i a b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)+\frac{1}{4} i a b^2 c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{4} i a b^2 c^3 \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{2} i a b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} i a b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{4} i b^2 c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) \text{Li}_2\left (1-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (1+\frac{i x}{c}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )+\frac{1}{2} \left (i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac{1}{2} i a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2-\frac{3}{4} i a b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{16} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{24} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a^2 b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{2} a b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i a b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{8} i b^3 c^2 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{16} b^3 c x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} a b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{24} i b^3 x^3 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} a^2 b c^3 \log (c-i x)+\frac{1}{4} i a b^2 c^3 \log (c-i x)-\frac{1}{2} i a b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{4} i a b^2 c^3 \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{2} i a b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{8} b^3 c^3 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{1}{8} b c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} i a b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{2} i a b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{4} i b^2 c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) \text{Li}_2\left (1-\frac{i c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (\frac{c-i x}{2 c}\right )-\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (\frac{c+i x}{2 c}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{2} i a b^2 c^3 \text{Li}_2\left (1+\frac{i x}{c}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{i c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log ^2\left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i b^3\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.678251, size = 330, normalized size = 1.44 \[ a b^2 \left (-i c^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+\left (x^3-i c^3\right ) \tan ^{-1}\left (\frac{c}{x}\right )^2+c \tan ^{-1}\left (\frac{c}{x}\right ) \left (2 c^2 \log \left (1-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+c^2+x^2\right )+c^2 x\right )+\frac{1}{24} b^3 \left (24 i c^3 \tan ^{-1}\left (\frac{c}{x}\right ) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+12 c^3 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )-24 c^3 \log \left (\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right )+8 i c^3 \tan ^{-1}\left (\frac{c}{x}\right )^3+12 c^3 \tan ^{-1}\left (\frac{c}{x}\right )^2+24 c^2 x \tan ^{-1}\left (\frac{c}{x}\right )+24 c^3 \tan ^{-1}\left (\frac{c}{x}\right )^2 \log \left (1-e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )-i \pi ^3 c^3+12 c x^2 \tan ^{-1}\left (\frac{c}{x}\right )^2+8 x^3 \tan ^{-1}\left (\frac{c}{x}\right )^3\right )-\frac{1}{2} a^2 b c^3 \log \left (c^2+x^2\right )+\frac{1}{2} a^2 b c x^2+a^2 b x^3 \tan ^{-1}\left (\frac{c}{x}\right )+\frac{a^3 x^3}{3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcTan[c/x])^3,x]

[Out]

(a^2*b*c*x^2)/2 + (a^3*x^3)/3 + a^2*b*x^3*ArcTan[c/x] - (a^2*b*c^3*Log[c^2 + x^2])/2 + a*b^2*(c^2*x + ((-I)*c^
3 + x^3)*ArcTan[c/x]^2 + c*ArcTan[c/x]*(c^2 + x^2 + 2*c^2*Log[1 - E^((2*I)*ArcTan[c/x])]) - I*c^3*PolyLog[2, E
^((2*I)*ArcTan[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTan[c/x] + 12*c^3*ArcTan[c/x]^2 + 12*c*x^2*ArcTan[c
/x]^2 + (8*I)*c^3*ArcTan[c/x]^3 + 8*x^3*ArcTan[c/x]^3 + 24*c^3*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcTan[c/x])] -
24*c^3*Log[c/(Sqrt[1 + c^2/x^2]*x)] + (24*I)*c^3*ArcTan[c/x]*PolyLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c^3*Poly
Log[3, E^((-2*I)*ArcTan[c/x])]))/24

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Maple [C]  time = 1.027, size = 6441, normalized size = 28.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{24} \, b^{3} x^{3} \arctan \left (c, x\right )^{3} - \frac{1}{32} \, b^{3} x^{3} \arctan \left (c, x\right ) \log \left (c^{2} + x^{2}\right )^{2} + \frac{1}{3} \, a^{3} x^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \arctan \left (\frac{c}{x}\right ) -{\left (c^{2} \log \left (c^{2} + x^{2}\right ) - x^{2}\right )} c\right )} a^{2} b + \int \frac{4 \, b^{3} c x^{3} \arctan \left (c, x\right )^{2} + 4 \, b^{3} x^{4} \arctan \left (c, x\right ) \log \left (c^{2} + x^{2}\right ) + 4 \,{\left (7 \, b^{3} \arctan \left (c, x\right )^{3} + 24 \, a b^{2} \arctan \left (c, x\right )^{2}\right )} x^{4} + 4 \,{\left (7 \, b^{3} c^{2} \arctan \left (c, x\right )^{3} + 24 \, a b^{2} c^{2} \arctan \left (c, x\right )^{2}\right )} x^{2} +{\left (3 \, b^{3} c^{2} x^{2} \arctan \left (c, x\right ) + 3 \, b^{3} x^{4} \arctan \left (c, x\right ) - b^{3} c x^{3}\right )} \log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctan(c/x))^3,x, algorithm="maxima")

[Out]

1/24*b^3*x^3*arctan2(c, x)^3 - 1/32*b^3*x^3*arctan2(c, x)*log(c^2 + x^2)^2 + 1/3*a^3*x^3 + 1/2*(2*x^3*arctan(c
/x) - (c^2*log(c^2 + x^2) - x^2)*c)*a^2*b + integrate(1/32*(4*b^3*c*x^3*arctan2(c, x)^2 + 4*b^3*x^4*arctan2(c,
x)*log(c^2 + x^2) + 4*(7*b^3*arctan2(c, x)^3 + 24*a*b^2*arctan2(c, x)^2)*x^4 + 4*(7*b^3*c^2*arctan2(c, x)^3 +
24*a*b^2*c^2*arctan2(c, x)^2)*x^2 + (3*b^3*c^2*x^2*arctan2(c, x) + 3*b^3*x^4*arctan2(c, x) - b^3*c*x^3)*log(c
^2 + x^2)^2)/(c^2 + x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctan(c/x))^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atan(c/x))**3,x)

[Out]

Integral(x**2*(a + b*atan(c/x))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctan(c/x))^3,x, algorithm="giac")

[Out]

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