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

Optimal. Leaf size=119 \[ 3 i b^2 c \text{PolyLog}\left (2,1-\frac{2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )-\frac{3}{2} b^3 c \text{PolyLog}\left (3,1-\frac{2 c}{c+i x}\right )+i c \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3+x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^3-3 b c \log \left (\frac{2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2 \]

[Out]

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

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Rubi [F]  time = 0.738651, 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 = {} \[ \int \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTan[c/x])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.256732, size = 215, normalized size = 1.81 \[ -3 a b^2 \left (-i c \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )-(x+i c) \tan ^{-1}\left (\frac{c}{x}\right )^2+2 c \tan ^{-1}\left (\frac{c}{x}\right ) \log \left (1-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )\right )-\frac{1}{8} b^3 \left (24 i c \tan ^{-1}\left (\frac{c}{x}\right ) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+12 c \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+8 i c \tan ^{-1}\left (\frac{c}{x}\right )^3-8 x \tan ^{-1}\left (\frac{c}{x}\right )^3+24 c \tan ^{-1}\left (\frac{c}{x}\right )^2 \log \left (1-e^{-2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )-i \pi ^3 c\right )+\frac{3}{2} a^2 b c \log \left (c^2+x^2\right )+3 a^2 b x \tan ^{-1}\left (\frac{c}{x}\right )+a^3 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c/x])^3,x]

[Out]

a^3*x + 3*a^2*b*x*ArcTan[c/x] + (3*a^2*b*c*Log[c^2 + x^2])/2 - 3*a*b^2*(-((I*c + x)*ArcTan[c/x]^2) + 2*c*ArcTa
n[c/x]*Log[1 - E^((2*I)*ArcTan[c/x])] - I*c*PolyLog[2, E^((2*I)*ArcTan[c/x])]) - (b^3*((-I)*c*Pi^3 + (8*I)*c*A
rcTan[c/x]^3 - 8*x*ArcTan[c/x]^3 + 24*c*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcTan[c/x])] + (24*I)*c*ArcTan[c/x]*P
olyLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c*PolyLog[3, E^((-2*I)*ArcTan[c/x])]))/8

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Maple [C]  time = 0.33, size = 2363, normalized size = 19.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*I*c*b^3*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1))*csgn(I/((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(I*((1+I*c/x)^2/(1
+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))*arctan(c/x)^2+3/4*I*c*b^3*arctan(c/x)^2*Pi*csgn(I/((1+I*c/x)^2/(1+c^
2/x^2)+1)^2)*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)+x*a
^3+b^3*x*arctan(c/x)^3-6*c*b^3*polylog(3,(1+I*c/x)/(1+c^2/x^2)^(1/2))-6*c*b^3*polylog(3,-(1+I*c/x)/(1+c^2/x^2)
^(1/2))-3*c*b^3*arctan(c/x)^2*ln(2)+3*a^2*b*x*arctan(c/x)+3*a*b^2*x*arctan(c/x)^2-3*c*b^3*arctan(c/x)^2*ln(1+(
1+I*c/x)/(1+c^2/x^2)^(1/2))-3*c*b^3*arctan(c/x)^2*ln(1-(1+I*c/x)/(1+c^2/x^2)^(1/2))+3/2*c*b^3*arctan(c/x)^2*ln
(1+c^2/x^2)-3*c*b^3*ln(c/x)*arctan(c/x)^2-3*c*b^3*arctan(c/x)^2*ln((1+I*c/x)/(1+c^2/x^2)^(1/2))+3*c*b^3*arctan
(c/x)^2*ln((1+I*c/x)^2/(1+c^2/x^2)-1)+I*c*b^3*arctan(c/x)^3-3*c*a^2*b*ln(c/x)+3/2*c*a^2*b*ln(1+c^2/x^2)+6*I*c*
b^3*arctan(c/x)*polylog(2,-(1+I*c/x)/(1+c^2/x^2)^(1/2))+6*I*c*b^3*arctan(c/x)*polylog(2,(1+I*c/x)/(1+c^2/x^2)^
(1/2))-3/2*I*c*b^3*Pi*arctan(c/x)^2+3/4*I*c*a*b^2*ln(c/x+I)^2+3/2*I*c*a*b^2*dilog(1/2*I*(c/x-I))-3*I*c*a*b^2*d
ilog(1+I*c/x)+3*I*c*a*b^2*dilog(1-I*c/x)-3/4*I*c*a*b^2*ln(c/x-I)^2-3/2*I*c*a*b^2*dilog(-1/2*I*(c/x+I))+3*c*a*b
^2*arctan(c/x)*ln(1+c^2/x^2)-6*c*a*b^2*arctan(c/x)*ln(c/x)-3/2*I*c*b^3*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/(
(1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))*arctan(c/x)^2+3/2*I*
c*b^3*Pi*csgn(I/((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2
*arctan(c/x)^2+3/2*I*c*b^3*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1))*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x
)^2/(1+c^2/x^2)+1))^2*arctan(c/x)^2+3/2*I*c*b^3*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)
+1))*csgn(((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2*arctan(c/x)^2-3/4*I*c*b^3*arctan(c/x)^2*P
i*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1))^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2)+3/4*I*c*b^3*arctan(c/x)^2*Pi*csg
n(I*(1+I*c/x)/(1+c^2/x^2)^(1/2))^2*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))-3/4*I*c*b^3*arctan(c/x)^2*Pi*csgn(I/((1+I*c
/x)^2/(1+c^2/x^2)+1)^2)*csgn(I*(1+I*c/x)^2/(1+c^2/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^2-3/4*I*c*b^3*arctan(c/x
)^2*Pi*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^2-3/2*I*c
*b^3*arctan(c/x)^2*Pi*csgn(I*(1+I*c/x)/(1+c^2/x^2)^(1/2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))^2+3/2*I*c*b^3*arctan
(c/x)^2*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^2+3/2*I*c*a*b^2*ln(c/x+I)
*ln(1/2*I*(c/x-I))-3*I*c*a*b^2*ln(c/x)*ln(1+I*c/x)+3*I*c*a*b^2*ln(c/x)*ln(1-I*c/x)+3/2*I*c*b^3*Pi*csgn(((1+I*c
/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2*arctan(c/x)^2-3/2*I*c*b^3*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^
2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^3*arctan(c/x)^2-3/4*I*c*b^3*arctan(c/x)^2*Pi*csgn(I*((1+I*c/x)^2/(1+c^2/x^2
)+1)^2)^3-3/2*I*c*b^3*Pi*csgn(((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^3*arctan(c/x)^2+3/4*I*c
*b^3*arctan(c/x)^2*Pi*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))^3+3/4*I*c*b^3*arctan(c/x)^2*Pi*csgn(I*(1+I*c/x)^2/(1+c^2
/x^2)/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)^3+3/2*I*c*a*b^2*ln(1+c^2/x^2)*ln(c/x-I)-3/2*I*c*a*b^2*ln(c/x-I)*ln(-1/2*I
*(c/x+I))-3/2*I*c*a*b^2*ln(1+c^2/x^2)*ln(c/x+I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{7}{8} \, b^{3} c \arctan \left (\frac{c}{x}\right )^{3} \arctan \left (\frac{x}{c}\right ) + 3 \, a b^{2} c \arctan \left (\frac{c}{x}\right )^{2} \arctan \left (\frac{x}{c}\right ) + \frac{1}{8} \, b^{3} x \arctan \left (c, x\right )^{3} - \frac{3}{32} \, b^{3} x \arctan \left (c, x\right ) \log \left (c^{2} + x^{2}\right )^{2} +{\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 )} a b^{2} c^{2} + \frac{7}{32} \,{\left (\frac{6 \, \arctan \left (\frac{c}{x}\right )^{2} \arctan \left (\frac{x}{c}\right )^{2}}{c} + \frac{4 \, \arctan \left (\frac{c}{x}\right ) \arctan \left (\frac{x}{c}\right )^{3}}{c} + \frac{\arctan \left (\frac{x}{c}\right )^{4}}{c}\right )} b^{3} c^{2} + 3 \, b^{3} c^{2} \int \frac{\arctan \left (\frac{c}{x}\right ) \log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 12 \, b^{3} c \int \frac{x \arctan \left (\frac{c}{x}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} - 3 \, b^{3} c \int \frac{x \log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + \frac{3}{2} \,{\left (2 \, x \arctan \left (\frac{c}{x}\right ) + c \log \left (c^{2} + x^{2}\right )\right )} a^{2} b + a^{3} x + 28 \, b^{3} \int \frac{x^{2} \arctan \left (\frac{c}{x}\right )^{3}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 3 \, b^{3} \int \frac{x^{2} \arctan \left (\frac{c}{x}\right ) \log \left (c^{2} + x^{2}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 96 \, a b^{2} \int \frac{x^{2} \arctan \left (\frac{c}{x}\right )^{2}}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} + 12 \, b^{3} \int \frac{x^{2} \arctan \left (\frac{c}{x}\right ) \log \left (c^{2} + x^{2}\right )}{32 \,{\left (c^{2} + x^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/8*b^3*c*arctan(c/x)^3*arctan(x/c) + 3*a*b^2*c*arctan(c/x)^2*arctan(x/c) + 1/8*b^3*x*arctan2(c, x)^3 - 3/32*b
^3*x*arctan2(c, x)*log(c^2 + x^2)^2 + (3*arctan(c/x)*arctan(x/c)^2/c + arctan(x/c)^3/c)*a*b^2*c^2 + 7/32*(6*ar
ctan(c/x)^2*arctan(x/c)^2/c + 4*arctan(c/x)*arctan(x/c)^3/c + arctan(x/c)^4/c)*b^3*c^2 + 3*b^3*c^2*integrate(1
/32*arctan(c/x)*log(c^2 + x^2)^2/(c^2 + x^2), x) + 12*b^3*c*integrate(1/32*x*arctan(c/x)^2/(c^2 + x^2), x) - 3
*b^3*c*integrate(1/32*x*log(c^2 + x^2)^2/(c^2 + x^2), x) + 3/2*(2*x*arctan(c/x) + c*log(c^2 + x^2))*a^2*b + a^
3*x + 28*b^3*integrate(1/32*x^2*arctan(c/x)^3/(c^2 + x^2), x) + 3*b^3*integrate(1/32*x^2*arctan(c/x)*log(c^2 +
x^2)^2/(c^2 + x^2), x) + 96*a*b^2*integrate(1/32*x^2*arctan(c/x)^2/(c^2 + x^2), x) + 12*b^3*integrate(1/32*x^
2*arctan(c/x)*log(c^2 + x^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} \arctan \left (\frac{c}{x}\right )^{3} + 3 \, a b^{2} \arctan \left (\frac{c}{x}\right )^{2} + 3 \, a^{2} b \arctan \left (\frac{c}{x}\right ) + a^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c/x) + a)^3, x)