Optimal. Leaf size=214 \[ 2 b^2 c^4 \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-i b c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {3}{4} b c^3 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-i b^3 c^4 \text {Li}_2\left (\frac {2}{1-\frac {i c}{x}}-1\right )+\frac {1}{4} b^3 c^4 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 x \]
[Out]
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Rubi [F] time = 4.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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]
[Out]
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*}
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Mathematica [A] time = 0.65, size = 253, normalized size = 1.18 \[ \frac {1}{4} \left (a^3 x^4+b \tan ^{-1}\left (\frac {c}{x}\right ) \left (3 a^2 \left (x^4-c^4\right )+2 a b c x \left (x^2-3 c^2\right )+8 b^2 c^4 \log \left (1-e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )+b^2 c^2 \left (c^2+x^2\right )\right )-3 a^2 b c^3 x+a^2 b c x^3+a b^2 c^4+a b^2 c^2 x^2+8 a b^2 c^4 \log \left (\frac {c}{x \sqrt {\frac {c^2}{x^2}+1}}\right )+b^2 \tan ^{-1}\left (\frac {c}{x}\right )^2 \left (3 a \left (x^4-c^4\right )+b c \left (-4 i c^3-3 c^2 x+x^3\right )\right )-4 i b^3 c^4 \text {Li}_2\left (e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )+b^3 \left (x^4-c^4\right ) \tan ^{-1}\left (\frac {c}{x}\right )^3+b^3 c^3 x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \arctan \left (\frac {c}{x}\right )^{3} + 3 \, a b^{2} x^{3} \arctan \left (\frac {c}{x}\right )^{2} + 3 \, a^{2} b x^{3} \arctan \left (\frac {c}{x}\right ) + a^{3} x^{3}, x\right ) \]
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 \[ \int {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 608, normalized size = 2.84 \[ -\frac {c^{4} b^{3} \arctan \left (\frac {c}{x}\right )^{3}}{4}-\frac {c^{4} b^{3} \arctan \left (\frac {x}{c}\right )}{4}+\frac {b^{3} x^{4} \arctan \left (\frac {c}{x}\right )^{3}}{4}+\frac {b^{3} c^{3} x}{4}+2 c^{4} a \,b^{2} \ln \left (\frac {c}{x}\right )-\frac {3 c^{4} a \,b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{4}+\frac {3 a \,b^{2} x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{4}+\frac {3 a^{2} b \,x^{4} \arctan \left (\frac {c}{x}\right )}{4}-\frac {3 c^{3} x \,a^{2} b}{4}+\frac {c \,b^{3} \arctan \left (\frac {c}{x}\right )^{2} x^{3}}{4}+\frac {c^{2} b^{3} \arctan \left (\frac {c}{x}\right ) x^{2}}{4}+\frac {i c^{4} b^{3} \dilog \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{2}-i c^{4} b^{3} \dilog \left (1-\frac {i c}{x}\right )+\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right )^{2}}{4}-\frac {i c^{4} b^{3} \ln \left (i+\frac {c}{x}\right )^{2}}{4}-\frac {i c^{4} b^{3} \dilog \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}+i c^{4} b^{3} \dilog \left (1+\frac {i c}{x}\right )+\frac {c^{2} x^{2} a \,b^{2}}{4}-c^{4} b^{3} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+2 c^{4} b^{3} \arctan \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}\right )+\frac {3 c^{4} a^{2} b \arctan \left (\frac {x}{c}\right )}{4}-\frac {i c^{4} b^{3} \ln \left (i+\frac {c}{x}\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}-i c^{4} b^{3} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )+\frac {i c^{4} b^{3} \ln \left (i+\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{2}+i c^{4} b^{3} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )-\frac {i c^{4} b^{3} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {c a \,b^{2} \arctan \left (\frac {c}{x}\right ) x^{3}}{2}-\frac {3 c^{3} a \,b^{2} x \arctan \left (\frac {c}{x}\right )}{2}+\frac {x^{4} a^{3}}{4}-c^{4} a \,b^{2} \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {3 c^{3} b^{3} \arctan \left (\frac {c}{x}\right )^{2} x}{4}+\frac {c \,a^{2} b \,x^{3}}{4} \]
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 \[ \frac {3}{4} \, a b^{2} x^{4} \arctan \left (\frac {c}{x}\right )^{2} + \frac {1}{4} \, a^{3} x^{4} + \frac {1}{4} \, {\left (3 \, x^{4} \arctan \left (\frac {c}{x}\right ) + {\left (3 \, c^{3} \arctan \left (\frac {x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c\right )} a^{2} b + \frac {1}{4} \, {\left ({\left (3 \, c^{2} \arctan \left (\frac {x}{c}\right )^{2} - 4 \, c^{2} \log \left (c^{2} + x^{2}\right ) + x^{2}\right )} c^{2} + 2 \, {\left (3 \, c^{3} \arctan \left (\frac {x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c \arctan \left (\frac {c}{x}\right )\right )} a b^{2} - \frac {1}{64} \, {\left (12 \, c^{4} \arctan \left (\frac {c}{x}\right )^{2} \arctan \left (\frac {x}{c}\right ) + 8 \, c^{4} \arctan \left (c, x\right )^{3} - 8 \, x^{4} \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^{5} + 12 \, c^{3} x \arctan \left (c, x\right )^{2} - 4 \, c x^{3} \arctan \left (c, x\right )^{2} + 192 \, c^{5} \int \frac {\log \left (c^{2} + x^{2}\right )^{2}}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} + 1536 \, c^{4} \int \frac {x \arctan \left (\frac {c}{x}\right )}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} + 768 \, c^{3} \int \frac {x^{2} \log \left (c^{2} + x^{2}\right )}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - 2048 \, c^{2} \int \frac {x^{3} \arctan \left (\frac {c}{x}\right )^{3}}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - 512 \, c^{2} \int \frac {x^{3} \arctan \left (\frac {c}{x}\right )}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - {\left (3 \, c^{3} x - c x^{3}\right )} \log \left (c^{2} + x^{2}\right )^{2} - 768 \, c \int \frac {x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - 192 \, c \int \frac {x^{4} \log \left (c^{2} + x^{2}\right )^{2}}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - 256 \, c \int \frac {x^{4} \log \left (c^{2} + x^{2}\right )}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - 2048 \, \int \frac {x^{5} \arctan \left (\frac {c}{x}\right )^{3}}{64 \, {\left (c^{2} + x^{2}\right )}}\,{d x}\right )} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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