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

Optimal. Leaf size=214 \[ -i b^3 c^4 \text{PolyLog}\left (2,-1+\frac{2}{1-\frac{i c}{x}}\right )+\frac{1}{4} b^2 c^2 x^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+2 b^2 c^4 \log \left (2-\frac{2}{1-\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )-\frac{1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3-i b c^4 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{3}{4} b c^3 x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{4} b^3 c^3 x+\frac{1}{4} b^3 c^4 \cot ^{-1}\left (\frac{x}{c}\right ) \]

[Out]

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

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Rubi [F]  time = 4.62858, 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 = {} \[ \int x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*(a + b*ArcTan[c/x])^3,x]

[Out]

(-3*a^2*b*c^3*x)/8 - ((5*I)/16)*a*b^2*c^3*x + (b^3*c^3*x)/16 - ((3*I)/16)*a^2*b*c^2*x^2 + (3*a*b^2*c^2*x^2)/16
+ (a^2*b*c*x^3)/8 - (11*a*b^2*c^4*Log[I - c/x])/16 - (I/32)*b^3*c^4*Log[I - c/x] - ((3*I)/8)*a*b^2*c^3*x*Log[
1 - (I*c)/x] + (3*a*b^2*c^2*x^2*Log[1 - (I*c)/x])/16 + (I/8)*a*b^2*c*x^3*Log[1 - (I*c)/x] + ((5*I)/32)*b^2*c^3
*(1 - (I*c)/x)*x*(2*a + I*b*Log[1 - (I*c)/x]) + (b^2*c^2*x^2*(2*a + I*b*Log[1 - (I*c)/x]))/32 + ((5*I)/64)*b*c
^4*(2*a + I*b*Log[1 - (I*c)/x])^2 - (3*b*c^3*(1 - (I*c)/x)*x*(2*a + I*b*Log[1 - (I*c)/x])^2)/32 + ((3*I)/64)*b
*c^2*x^2*(2*a + I*b*Log[1 - (I*c)/x])^2 + (b*c*x^3*(2*a + I*b*Log[1 - (I*c)/x])^2)/32 - (c^4*(2*a + I*b*Log[1
- (I*c)/x])^3)/32 + (x^4*(2*a + I*b*Log[1 - (I*c)/x])^3)/32 + ((3*I)/4)*a*b^2*c^3*x*Log[1 + (I*c)/x] - (5*b^3*
c^3*(1 + (I*c)/x)*x*Log[1 + (I*c)/x])/32 - (I/32)*b^3*c^2*x^2*Log[1 + (I*c)/x] - (I/4)*a*b^2*c*x^3*Log[1 + (I*
c)/x] - ((3*I)/8)*a^2*b*x^4*Log[1 + (I*c)/x] + (3*a*b^2*x^4*Log[1 - (I*c)/x]*Log[1 + (I*c)/x])/8 + (3*a*b^2*c^
4*Log[1 + (I*c)/x]^2)/16 + ((5*I)/64)*b^3*c^4*Log[1 + (I*c)/x]^2 + (3*b^3*c^3*(1 + (I*c)/x)*x*Log[1 + (I*c)/x]
^2)/32 + ((3*I)/64)*b^3*c^2*x^2*Log[1 + (I*c)/x]^2 - (b^3*c*x^3*Log[1 + (I*c)/x]^2)/32 - (3*a*b^2*x^4*Log[1 +
(I*c)/x]^2)/16 - (I/32)*b^3*c^4*Log[1 + (I*c)/x]^3 + (I/32)*b^3*x^4*Log[1 + (I*c)/x]^3 + (I/32)*b^3*c^4*Log[I
+ c/x] + ((3*I)/8)*a^2*b*c^4*Log[c - I*x] - (5*a*b^2*c^4*Log[c - I*x])/16 - (3*a*b^2*c^4*Log[1 - (I*c)/x]*Log[
c - I*x])/8 - (5*a*b^2*c^4*Log[c + I*x])/16 - (3*a*b^2*c^4*Log[1 + (I*c)/x]*Log[c + I*x])/8 + (3*a*b^2*c^4*Log
[(c - I*x)/(2*c)]*Log[c + I*x])/8 + (3*a*b^2*c^4*Log[c - I*x]*Log[(c + I*x)/(2*c)])/8 + ((3*I)/32)*b^3*c^4*Log
[1 + (I*c)/x]^2*Log[((-I)*c)/x] + ((3*I)/32)*b*c^4*(2*a + I*b*Log[1 - (I*c)/x])^2*Log[(I*c)/x] - (11*a*b^2*c^4
*Log[x])/8 - (3*a*b^2*c^4*Log[c + I*x]*Log[((-I)*x)/c])/8 - (3*a*b^2*c^4*Log[c - I*x]*Log[(I*x)/c])/8 - (3*b^2
*c^4*(2*a + I*b*Log[1 - (I*c)/x])*PolyLog[2, 1 - (I*c)/x])/16 + ((3*I)/16)*b^3*c^4*Log[1 + (I*c)/x]*PolyLog[2,
1 + (I*c)/x] + (3*a*b^2*c^4*PolyLog[2, (c - I*x)/(2*c)])/8 + (3*a*b^2*c^4*PolyLog[2, (c + I*x)/(2*c)])/8 + (3
*a*b^2*c^4*PolyLog[2, ((-I)*c)/x])/8 + ((11*I)/32)*b^3*c^4*PolyLog[2, ((-I)*c)/x] - ((11*I)/32)*b^3*c^4*PolyLo
g[2, (I*c)/x] - (3*a*b^2*c^4*PolyLog[2, 1 - (I*x)/c])/8 - (3*a*b^2*c^4*PolyLog[2, 1 + (I*x)/c])/8 + ((3*I)/16)
*b^3*c^4*PolyLog[3, 1 - (I*c)/x] - ((3*I)/16)*b^3*c^4*PolyLog[3, 1 + (I*c)/x] + ((3*I)/8)*b^3*Defer[Int][x^3*L
og[1 - (I*c)/x]^2*Log[1 + (I*c)/x], x] - ((3*I)/8)*b^3*Defer[Int][x^3*Log[1 - (I*c)/x]*Log[1 + (I*c)/x]^2, x]

Rubi steps

\begin{align*} \int x^3 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{3}{8} i b x^3 \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^3 \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^3 \log ^3\left (1+\frac{i c}{x}\right )\right ) \, dx\\ &=\frac{1}{8} \int x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3 \, dx+\frac{1}{8} (3 i b) \int x^3 \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^3 \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^3 \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^5} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} (3 i b) \int \left (-4 a^2 x^3 \log \left (1+\frac{i c}{x}\right )-4 i a b x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+b^2 x^3 \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^3 \log ^2\left (1+\frac{i c}{x}\right )+b x^3 \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^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{2} \left (3 i a^2 b\right ) \int x^3 \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{4} \left (3 a b^2\right ) \int x^3 \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} \left (3 a b^2\right ) \int x^3 \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^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{32} (3 b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^4 (1-i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{32} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^4 (1+i c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{32} (3 i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^4} \, 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^5} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^3 \log \left (1-\frac{i c}{x}\right )}{4 (-c+i x)} \, dx-\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^3 \log \left (1+\frac{i c}{x}\right )}{4 (-c-i x)} \, dx-\frac{1}{32} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^4} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 a^2 b c\right ) \int \frac{x^2}{1+\frac{i c}{x}} \, dx\\ &=\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )-\frac{1}{32} (3 i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^4} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{32} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^4} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{32} (3 b c) \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}{8} \left (3 a^2 b c\right ) \int \frac{x^3}{i c+x} \, dx+\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^4 (1+i c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 a b^2 c\right ) \int \frac{x^3 \log \left (1-\frac{i c}{x}\right )}{-c+i x} \, dx-\frac{1}{8} \left (3 a b^2 c\right ) \int \frac{x^3 \log \left (1+\frac{i c}{x}\right )}{-c-i x} \, dx-\frac{1}{32} \left (3 b^3 c\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}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{32} (3 b c) \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 (3 a^2 b c\right ) \int \left (-c^2+\frac{c^3}{c-i x}-i c x+x^2\right ) \, dx-\frac{1}{16} \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 )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x^4}-\frac{i c \log (1+i c x)}{x^3}-\frac{c^2 \log (1+i c x)}{x^2}+\frac{i c^3 \log (1+i c x)}{x}-\frac{i c^4 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 a b^2 c\right ) \int \left (i c^2 \log \left (1-\frac{i c}{x}\right )-\frac{i c^3 \log \left (1-\frac{i c}{x}\right )}{c-i x}-c x \log \left (1-\frac{i c}{x}\right )-i x^2 \log \left (1-\frac{i c}{x}\right )\right ) \, dx-\frac{1}{8} \left (3 a b^2 c\right ) \int \left (-i c^2 \log \left (1+\frac{i c}{x}\right )+\frac{i c^3 \log \left (1+\frac{i c}{x}\right )}{c+i x}-c x \log \left (1+\frac{i c}{x}\right )+i x^2 \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^3} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{32} \left (3 b^3 c\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}{32} \left (3 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 )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{32} \left (3 i b^3 c^2\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{3}{8} a^2 b c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{1}{8} a^2 b c x^3+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{16} \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 )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (3 i a b^2 c\right ) \int x^2 \log \left (1-\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i a b^2 c\right ) \int x^2 \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 i a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^4} \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^3} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{32} \left (3 i b c^2\right ) \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}{16} \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 )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{32} \left (3 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 )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3} \, dx,x,\frac{1}{x}\right )-\frac{1}{16} \left (i b^3 c^2\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}{32} \left (3 i b^3 c^2\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}{32} \left (3 i b^3 c^2\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}{32} \left (3 b c^3\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 (3 i a b^2 c^3\right ) \int \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (3 i a b^2 c^3\right ) \int \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2} \, dx,x,\frac{1}{x}\right )+\frac{1}{32} \left (3 b^3 c^3\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}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx-\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 a b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{16} \left (b^2 c^2\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}{32} \left (3 b^2 c^2\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}{8} \left (a b^2 c^2\right ) \int \frac{x}{1-\frac{i c}{x}} \, dx+\frac{1}{8} \left (a b^2 c^2\right ) \int \frac{x}{1+\frac{i c}{x}} \, dx-\frac{1}{8} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{32} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{32} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{16} \left (i b^3 c^2\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}{32} \left (3 i b^3 c^2\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}{32} \left (3 b c^3\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}{16} \left (i b^2 c^3\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}{32} \left (3 i b^2 c^3\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}{16} \left (3 i b^2 c^3\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}{16} \left (3 i a b^2 c^3\right ) \int \frac{1}{1-\frac{i c}{x}} \, dx+\frac{1}{16} \left (3 i a b^2 c^3\right ) \int \frac{1}{1+\frac{i c}{x}} \, dx+\frac{1}{16} \left (3 i a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+i c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{16} \left (b^3 c^3\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}{32} \left (3 b^3 c^3\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}{32} \left (3 b^3 c^3\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}{16} \left (3 b^3 c^3\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}{32} \left (3 i b c^4\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{8} \left (3 a b^2 c^4\right ) \int \frac{1}{\left (1-\frac{i c}{x}\right ) x} \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \int \frac{1}{\left (1+\frac{i c}{x}\right ) x} \, dx+\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{32} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i a b^2 c^5\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx-\frac{1}{8} \left (3 i a b^2 c^5\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{5}{32} i b^2 c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{5}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{32} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{3}{32} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )-\frac{3}{8} a b^2 c^4 \log (x)+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{3}{16} i b^3 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (a b^2 c^2\right ) \int \frac{x^2}{-i c+x} \, dx+\frac{1}{8} \left (a b^2 c^2\right ) \int \frac{x^2}{i c+x} \, dx-\frac{1}{8} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{i c}{x^2}-\frac{c^2}{x}+\frac{c^3}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{32} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{c^2}{(-1+x)^2}+\frac{c^2}{-1+x}-\frac{c^2}{x}\right ) \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{32} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{c^2}{(-1+x)^2}+\frac{c^2}{-1+x}-\frac{c^2}{x}\right ) \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{16} \left (i b^2 c^3\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}{32} \left (3 i b^2 c^3\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}{16} \left (3 i a b^2 c^3\right ) \int \frac{x}{-i c+x} \, dx+\frac{1}{16} \left (3 i a b^2 c^3\right ) \int \frac{x}{i c+x} \, dx+\frac{1}{16} \left (3 i a b^2 c^3\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}{16} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{16} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{16} \left (b^3 c^3\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}{32} \left (3 b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{32} \left (3 b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{32} \left (3 b^3 c^3\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}{16} \left (3 b^3 c^3\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}{32} \left (3 c^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{16} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{32} \left (3 b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{16} \left (3 b^2 c^4\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}{8} \left (3 a b^2 c^4\right ) \int \frac{1}{-i c+x} \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \int \frac{1}{i c+x} \, dx+\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{32} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log \left (1+\frac{i c}{x}\right )\right )+\frac{1}{32} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )-\frac{1}{16} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) \log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i a b^2 c^5\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx-\frac{1}{8} \left (3 i a b^2 c^5\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx-\frac{1}{8} \left (3 i a b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{5}{16} i a b^2 c^3 x+\frac{1}{16} b^3 c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{1}{16} a b^2 c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{11}{16} a b^2 c^4 \log \left (i-\frac{c}{x}\right )+\frac{1}{32} i b^3 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{5}{32} i b^2 c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{5}{64} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{32} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{32} i b^3 c^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{5}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{5}{64} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{3}{8} a b^2 c^4 \log (c+i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{32} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{3}{32} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )-\frac{11}{8} a b^2 c^4 \log (x)-\frac{3}{16} b^2 c^4 \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{3}{16} i b^3 c^4 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{11}{32} i b^3 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{3}{16} i b^3 c^4 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{8} \left (a b^2 c^2\right ) \int \left (-i c+\frac{i c^2}{c-i x}+x\right ) \, dx+\frac{1}{8} \left (a b^2 c^2\right ) \int \left (i c-\frac{i c^2}{c+i x}+x\right ) \, dx+\frac{1}{16} \left (3 i a b^2 c^3\right ) \int \left (1-\frac{c}{c-i x}\right ) \, dx-\frac{1}{16} \left (3 i a b^2 c^3\right ) \int \left (1-\frac{c}{c+i x}\right ) \, dx-\frac{1}{16} \left (b^3 c^3\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}{32} \left (3 b^3 c^3\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 (3 i a b^2 c^4\right ) \int \frac{\log (c-i x)}{c+i x} \, dx-\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log (c+i x)}{c-i x} \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \int \frac{\log (c-i x)}{x} \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \int \frac{\log (c+i x)}{x} \, dx+\frac{1}{16} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{16} \left (3 i b^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+\frac{i c}{x}\right )\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{5}{16} i a b^2 c^3 x+\frac{1}{16} b^3 c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{3}{16} a b^2 c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{11}{16} a b^2 c^4 \log \left (i-\frac{c}{x}\right )+\frac{1}{32} i b^3 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{5}{32} i b^2 c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{5}{64} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{32} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{32} i b^3 c^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{5}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{5}{64} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c+i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{3}{32} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{3}{32} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )-\frac{11}{8} a b^2 c^4 \log (x)-\frac{3}{8} a b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{3}{16} b^2 c^4 \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{3}{16} i b^3 c^4 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{11}{32} i b^3 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{11}{32} i b^3 c^4 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1-\frac{i c}{x}\right )-\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx+\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx+\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx-\frac{1}{8} \left (3 i a b^2 c^4\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{5}{16} i a b^2 c^3 x+\frac{1}{16} b^3 c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{3}{16} a b^2 c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{11}{16} a b^2 c^4 \log \left (i-\frac{c}{x}\right )+\frac{1}{32} i b^3 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{5}{32} i b^2 c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{5}{64} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{32} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{32} i b^3 c^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{5}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{5}{64} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c+i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{3}{32} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{3}{32} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )-\frac{11}{8} a b^2 c^4 \log (x)-\frac{3}{8} a b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{3}{16} b^2 c^4 \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{3}{16} i b^3 c^4 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{11}{32} i b^3 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{11}{32} i b^3 c^4 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{3}{8} a b^2 c^4 \text{Li}_2\left (1-\frac{i x}{c}\right )-\frac{3}{8} a b^2 c^4 \text{Li}_2\left (1+\frac{i x}{c}\right )+\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1-\frac{i c}{x}\right )-\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )-\frac{1}{8} \left (3 a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac{3}{8} a^2 b c^3 x-\frac{5}{16} i a b^2 c^3 x+\frac{1}{16} b^3 c^3 x-\frac{3}{16} i a^2 b c^2 x^2+\frac{3}{16} a b^2 c^2 x^2+\frac{1}{8} a^2 b c x^3-\frac{11}{16} a b^2 c^4 \log \left (i-\frac{c}{x}\right )+\frac{1}{32} i b^3 c^4 \log \left (1-\frac{i c}{x}\right )-\frac{3}{8} i a b^2 c^3 x \log \left (1-\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^2 x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{8} i a b^2 c x^3 \log \left (1-\frac{i c}{x}\right )+\frac{5}{32} i b^2 c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{32} b^2 c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{5}{64} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{3}{32} b c^3 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{3}{64} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{32} b c x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{32} c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3+\frac{1}{32} x^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^3-\frac{1}{32} i b^3 c^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{4} i a b^2 c^3 x \log \left (1+\frac{i c}{x}\right )-\frac{5}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log \left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^2 x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} i a b^2 c x^3 \log \left (1+\frac{i c}{x}\right )-\frac{3}{8} i a^2 b x^4 \log \left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 x^4 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{3}{16} a b^2 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{5}{64} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{32} b^3 c^3 \left (1+\frac{i c}{x}\right ) x \log ^2\left (1+\frac{i c}{x}\right )+\frac{3}{64} i b^3 c^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} b^3 c x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{3}{16} a b^2 x^4 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{32} i b^3 c^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{1}{32} i b^3 x^4 \log ^3\left (1+\frac{i c}{x}\right )+\frac{3}{8} i a^2 b c^4 \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c-i x)-\frac{3}{8} a b^2 c^4 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{5}{16} a b^2 c^4 \log (c+i x)-\frac{3}{8} a b^2 c^4 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{3}{32} i b^3 c^4 \log ^2\left (1+\frac{i c}{x}\right ) \log \left (-\frac{i c}{x}\right )+\frac{3}{32} i b c^4 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \log \left (\frac{i c}{x}\right )-\frac{11}{8} a b^2 c^4 \log (x)-\frac{3}{8} a b^2 c^4 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{3}{8} a b^2 c^4 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{3}{16} b^2 c^4 \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{3}{16} i b^3 c^4 \log \left (1+\frac{i c}{x}\right ) \text{Li}_2\left (1+\frac{i c}{x}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (\frac{c-i x}{2 c}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (\frac{c+i x}{2 c}\right )+\frac{3}{8} a b^2 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{11}{32} i b^3 c^4 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{11}{32} i b^3 c^4 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{3}{8} a b^2 c^4 \text{Li}_2\left (1-\frac{i x}{c}\right )-\frac{3}{8} a b^2 c^4 \text{Li}_2\left (1+\frac{i x}{c}\right )+\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1-\frac{i c}{x}\right )-\frac{3}{16} i b^3 c^4 \text{Li}_3\left (1+\frac{i c}{x}\right )+\frac{1}{8} \left (3 i b^3\right ) \int x^3 \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^3 \log \left (1-\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.590986, size = 253, normalized size = 1.18 \[ \frac{1}{4} \left (-4 i b^3 c^4 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+b \tan ^{-1}\left (\frac{c}{x}\right ) \left (3 a^2 \left (x^4-c^4\right )+2 a b c x \left (x^2-3 c^2\right )+b^2 c^2 \left (c^2+x^2\right )+8 b^2 c^4 \log \left (1-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )\right )-3 a^2 b c^3 x+a^2 b c x^3+a^3 x^4+a b^2 c^2 x^2+8 a b^2 c^4 \log \left (\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right )+b^2 \tan ^{-1}\left (\frac{c}{x}\right )^2 \left (3 a \left (x^4-c^4\right )+b c \left (-3 c^2 x-4 i c^3+x^3\right )\right )+a b^2 c^4+b^3 \left (x^4-c^4\right ) \tan ^{-1}\left (\frac{c}{x}\right )^3+b^3 c^3 x\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*b^2*c^4 - 3*a^2*b*c^3*x + b^3*c^3*x + a*b^2*c^2*x^2 + a^2*b*c*x^3 + a^3*x^4 + b^2*(b*c*((-4*I)*c^3 - 3*c^2*
x + x^3) + 3*a*(-c^4 + x^4))*ArcTan[c/x]^2 + b^3*(-c^4 + x^4)*ArcTan[c/x]^3 + b*ArcTan[c/x]*(2*a*b*c*x*(-3*c^2
+ x^2) + b^2*c^2*(c^2 + x^2) + 3*a^2*(-c^4 + x^4) + 8*b^2*c^4*Log[1 - E^((2*I)*ArcTan[c/x])]) + 8*a*b^2*c^4*L
og[c/(Sqrt[1 + c^2/x^2]*x)] - (4*I)*b^3*c^4*PolyLog[2, E^((2*I)*ArcTan[c/x])])/4

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Maple [B]  time = 0.108, size = 608, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a*b^2*x^4*arctan(c/x)^2 + 1/4*a^3*x^4 + 1/4*(3*x^4*arctan(c/x) + (3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c)*a^
2*b + 1/4*((3*c^2*arctan2(x, c)^2 - 4*c^2*log(c^2 + x^2) + x^2)*c^2 + 2*(3*c^3*arctan(x/c) - 3*c^2*x + x^3)*c*
arctan(c/x))*a*b^2 - 1/64*(12*c^4*arctan(c/x)^2*arctan(x/c) + 8*c^4*arctan2(c, x)^3 - 8*x^4*arctan2(c, x)^3 +
4*(3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/c)*c^5 + 12*c^3*x*arctan2(c, x)^2 - 4*c*x^3*arctan2(c, x)^2 +
192*c^5*integrate(1/64*log(c^2 + x^2)^2/(c^2 + x^2), x) + 1536*c^4*integrate(1/64*x*arctan(c/x)/(c^2 + x^2),
x) + 768*c^3*integrate(1/64*x^2*log(c^2 + x^2)/(c^2 + x^2), x) - 2048*c^2*integrate(1/64*x^3*arctan(c/x)^3/(c^
2 + x^2), x) - 512*c^2*integrate(1/64*x^3*arctan(c/x)/(c^2 + x^2), x) - (3*c^3*x - c*x^3)*log(c^2 + x^2)^2 - 7
68*c*integrate(1/64*x^4*arctan(c/x)^2/(c^2 + x^2), x) - 192*c*integrate(1/64*x^4*log(c^2 + x^2)^2/(c^2 + x^2),
x) - 256*c*integrate(1/64*x^4*log(c^2 + x^2)/(c^2 + x^2), x) - 2048*integrate(1/64*x^5*arctan(c/x)^3/(c^2 + x
^2), x))*b^3

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

[Out]

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

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

[Out]

Integral(x**3*(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^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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