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

Optimal. Leaf size=145 \[ \frac{3}{2} i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-\frac{i c}{x}}\right )-3 b^2 c^2 \log \left (2-\frac{2}{1-\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{3}{2} b c x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2 \]

[Out]

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

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Rubi [F]  time = 2.35191, 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 = {} \[ \int x \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(a + b*ArcTan[c/x])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.274443, size = 174, normalized size = 1.2 \[ \frac{1}{2} \left (3 i b^3 c^2 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (a x (a x+3 b c)-6 b^2 c^2 \log \left (\frac{c}{x \sqrt{\frac{c^2}{x^2}+1}}\right )\right )+3 b^2 \tan ^{-1}\left (\frac{c}{x}\right )^2 \left (a \left (c^2+x^2\right )+b c (x+i c)\right )+3 b \tan ^{-1}\left (\frac{c}{x}\right ) \left (a \left (a \left (c^2+x^2\right )+2 b c x\right )-2 b^2 c^2 \log \left (1-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )\right )+b^3 \left (c^2+x^2\right ) \tan ^{-1}\left (\frac{c}{x}\right )^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcTan[c/x])^3,x]

[Out]

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

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Maple [B]  time = 0.104, size = 507, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \, a b^{2} x^{2} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, a^{3} x^{2} + \frac{3}{2} \,{\left (x^{2} \arctan \left (\frac{c}{x}\right ) -{\left (c \arctan \left (\frac{x}{c}\right ) - x\right )} c\right )} a^{2} b - \frac{3}{2} \,{\left ({\left (\arctan \left (x, c\right )^{2} - \log \left (c^{2} + x^{2}\right )\right )} c^{2} + 2 \,{\left (c \arctan \left (\frac{x}{c}\right ) - x\right )} c \arctan \left (\frac{c}{x}\right )\right )} a b^{2} + \frac{1}{32} \,{\left (12 \, c^{2} \arctan \left (\frac{c}{x}\right )^{2} \arctan \left (\frac{x}{c}\right ) + 8 \, c^{2} \arctan \left (c, x\right )^{3} + 8 \, x^{2} \arctan \left (c, x\right )^{3} + 4 \,{\left (\frac{3 \, \arctan \left (\frac{c}{x}\right ) \arctan \left (\frac{x}{c}\right )^{2}}{c} + \frac{\arctan \left (\frac{x}{c}\right )^{3}}{c}\right )} c^{3} + 12 \, c x \arctan \left (c, x\right )^{2} + 96 \, c^{3} \int \frac{\log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} - 3 \, c x \log \left (c^{2} + x^{2}\right )^{2} + 512 \, c^{2} \int \frac{x \arctan \left (\frac{c}{x}\right )^{3}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 768 \, c^{2} \int \frac{x \arctan \left (\frac{c}{x}\right )}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 384 \, c \int \frac{x^{2} \arctan \left (\frac{c}{x}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 96 \, c \int \frac{x^{2} \log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 384 \, c \int \frac{x^{2} \log \left (c^{2} + x^{2}\right )}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 512 \, \int \frac{x^{3} \arctan \left (\frac{c}{x}\right )^{3}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*x^2*arctan(c/x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arctan(c/x) - (c*arctan(x/c) - x)*c)*a^2*b - 3/2*((arctan
2(x, c)^2 - log(c^2 + x^2))*c^2 + 2*(c*arctan(x/c) - x)*c*arctan(c/x))*a*b^2 + 1/32*(12*c^2*arctan(c/x)^2*arct
an(x/c) + 8*c^2*arctan2(c, x)^3 + 8*x^2*arctan2(c, x)^3 + 4*(3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/c)*
c^3 + 12*c*x*arctan2(c, x)^2 + 96*c^3*integrate(1/32*log(c^2 + x^2)^2/(c^2 + x^2), x) - 3*c*x*log(c^2 + x^2)^2
+ 512*c^2*integrate(1/32*x*arctan(c/x)^3/(c^2 + x^2), x) + 768*c^2*integrate(1/32*x*arctan(c/x)/(c^2 + x^2),
x) + 384*c*integrate(1/32*x^2*arctan(c/x)^2/(c^2 + x^2), x) + 96*c*integrate(1/32*x^2*log(c^2 + x^2)^2/(c^2 +
x^2), x) + 384*c*integrate(1/32*x^2*log(c^2 + x^2)/(c^2 + x^2), x) + 512*integrate(1/32*x^3*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 \arctan \left (\frac{c}{x}\right )^{3} + 3 \, a b^{2} x \arctan \left (\frac{c}{x}\right )^{2} + 3 \, a^{2} b x \arctan \left (\frac{c}{x}\right ) + a^{3} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c/x) + a)^3*x, x)