Optimal. Leaf size=108 \[ c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (\frac {2 c}{c-x}\right )-3 b^2 c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \text {PolyLog}\left (2,1-\frac {2 c}{c-x}\right )+\frac {3}{2} b^3 c \text {PolyLog}\left (3,1-\frac {2 c}{c-x}\right ) \]
[Out]
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Rubi [A]
time = 0.18, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6025, 6022,
6132, 6056, 6096, 6206, 6745} \begin {gather*} -3 b^2 c \text {Li}_2\left (1-\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \log \left (\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b^3 c \text {Li}_3\left (1-\frac {2 c}{c-x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6022
Rule 6025
Rule 6056
Rule 6096
Rule 6132
Rule 6206
Rule 6745
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx &=\int \left (a^3-\frac {3}{2} a^2 b \log \left (1-\frac {c}{x}\right )+\frac {3}{4} a b^2 \log ^2\left (1-\frac {c}{x}\right )-\frac {1}{8} b^3 \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} b^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 \log ^2\left (1+\frac {c}{x}\right )-\frac {3}{8} b^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 \log ^3\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=a^3 x-\frac {1}{2} \left (3 a^2 b\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2\right ) \int \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} b^3 \int \log ^3\left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} b^3 \int \log ^3\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c \log \left (1+\frac {c}{x}\right )}{-c+x} \, dx+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 c\right ) \int \frac {\log ^2\left (1-\frac {c}{x}\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 c\right ) \int \frac {\log ^2\left (1+\frac {c}{x}\right )}{x} \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{2} \left (3 a^2 b c\right ) \int \frac {1}{c+x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{-c+x} \, dx-\frac {1}{8} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1-c x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log ^2(1+c x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \frac {\log (-c-x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \frac {\log (-c+x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx-\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx,x,\frac {1}{x}\right )+\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \frac {\log (-c x) \log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (c \left (\frac {1}{c}-\frac {x}{c}\right )\right )}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-c \left (-\frac {1}{c}+\frac {x}{c}\right )\right )}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \left (-\frac {\log (-c-x)}{c (c-x)}-\frac {\log (-c-x)}{c x}\right ) \, dx+\frac {1}{2} \left (3 a b^2 c^2\right ) \int \left (\frac {\log (-c+x)}{c x}-\frac {\log (-c+x)}{c (c+x)}\right ) \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c-x)}{c-x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c-x)}{x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c+x)}{x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log (-c+x)}{c+x} \, dx+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{4} \left (3 b^3 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{-c-x} \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (\frac {x}{c}\right )}{-c+x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (-\frac {-c+x}{2 c}\right )}{-c-x} \, dx+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {\log \left (\frac {c+x}{2 c}\right )}{-c+x} \, dx\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c-x\right )+\frac {1}{2} \left (3 a b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c+x\right )\\ &=a^3 x-\frac {3}{2} a^2 b x \log \left (1-\frac {c}{x}\right )-\frac {3}{4} a b^2 (c-x) \log ^2\left (1-\frac {c}{x}\right )+\frac {1}{8} b^3 (c-x) \log ^3\left (1-\frac {c}{x}\right )+\frac {3}{2} a^2 b x \log \left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 x \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{4} a b^2 (c+x) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 (c+x) \log ^3\left (1+\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log \left (1-\frac {c}{x}\right ) \log (-c-x)+\frac {3}{2} a^2 b c \log (c-x)+\frac {3}{2} a b^2 c \log (-c-x) \log \left (\frac {c-x}{2 c}\right )-\frac {3}{8} b^3 c \log ^2\left (1-\frac {c}{x}\right ) \log \left (\frac {c}{x}\right )-\frac {3}{2} a b^2 c \log (-c-x) \log \left (-\frac {x}{c}\right )+\frac {3}{2} a b^2 c \log \left (1+\frac {c}{x}\right ) \log (-c+x)+\frac {3}{2} a b^2 c \log \left (\frac {x}{c}\right ) \log (-c+x)+\frac {3}{2} a^2 b c \log (c+x)-\frac {3}{2} a b^2 c \log (-c+x) \log \left (\frac {c+x}{2 c}\right )-\frac {3}{8} b^3 c \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{4} b^3 c \log \left (1-\frac {c}{x}\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c-x}{2 c}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (\frac {c+x}{2 c}\right )-\frac {3}{4} b^3 c \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{2} a b^2 c \text {Li}_2\left (1-\frac {x}{c}\right )-\frac {3}{2} a b^2 c \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (1-\frac {c}{x}\right )+\frac {3}{4} b^3 c \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 198, normalized size = 1.83 \begin {gather*} a^3 x+3 a^2 b x \tanh ^{-1}\left (\frac {c}{x}\right )+\frac {3}{2} a^2 b c \log \left (-c^2+x^2\right )-3 a b^2 \left (\tanh ^{-1}\left (\frac {c}{x}\right ) \left ((c-x) \tanh ^{-1}\left (\frac {c}{x}\right )+2 c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )-c \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{8} b^3 \left (-i c \pi ^3+8 c \tanh ^{-1}\left (\frac {c}{x}\right )^3+8 x \tanh ^{-1}\left (\frac {c}{x}\right )^3-24 c \tanh ^{-1}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )-24 c \tanh ^{-1}\left (\frac {c}{x}\right ) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+12 c \text {PolyLog}\left (3,e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.71, size = 1732, normalized size = 16.04
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1732\) |
default | \(\text {Expression too large to display}\) | \(1732\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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