Optimal. Leaf size=638 \[ \frac {43 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}+\frac {29 c^2 x^3 \sqrt {c+a^2 c x^2}}{1680}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1373 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{20160 a^3}-\frac {737 c^2 x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{10080 a}-\frac {83}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)+\frac {5 c^2 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{128 a^2}+\frac {59}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {17}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {5 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {397 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 5.82, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps
used = 238, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072,
5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \begin {gather*} -\frac {737 c^2 x^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{10080 a}+\frac {5 c^2 x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{128 a^2}+\frac {17}{48} a^2 c^2 x^5 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}-\frac {83}{840} a c^2 x^4 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}+\frac {59}{192} c^2 x^3 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}+\frac {43 c^2 x \sqrt {a^2 c x^2+c}}{4032 a^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {a^2 c x^2+c}+\frac {29 c^2 x^3 \sqrt {a^2 c x^2+c}}{1680}+\frac {1}{8} a^4 c^2 x^7 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {1373 c^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{20160 a^3}-\frac {1}{28} a^3 c^2 x^6 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}-\frac {397 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{5040 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 327
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rule 6724
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^3 \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (7 a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (a^5 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {3 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{48} \left (35 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{14} \left (3 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{24} \left (7 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (a^4 c^3\right ) \int \frac {x^6}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {7 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{64} \left (35 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a^2}-\frac {\left (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {1}{35} \left (6 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{30} \left (7 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{96} \left (35 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{168} \left (5 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{70} \left (3 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{120} \left (7 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c^2 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {13 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {\left (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}-\frac {359 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}+\frac {1}{224} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{280} \left (9 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{160} \left (7 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (2 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{90} \left (7 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{288} \left (35 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{12 a^2}+\frac {\left (8 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (35 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{128 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{3 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{4 a^2}+\frac {\left (4 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {\left (7 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}+\frac {\left (35 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{144 a}+\frac {\left (35 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{64 a}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{12 a^2}+\frac {\left (8 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{3 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{4 a^2}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{448 a^2}-\frac {\left (9 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{560 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{320 a^2}+\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{180 a^2}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{576 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{72 a^2}-\frac {\left (4 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (8 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{144 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{18 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{64 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{8 a^2}+\frac {\left (i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{72 a^2}+\frac {\left (8 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{18 a^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{8 a^2}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{128 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {379 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{448 a^2}-\frac {\left (9 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{560 a^2}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{320 a^2}-\frac {c^3 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{35 a^2}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{180 a^2}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{576 a^2}-\frac {\left (4 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{35 a^2}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{144 a^2}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{64 a^2}-\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (35 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 i c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1543\) vs. \(2(638)=1276\).
time = 3.58, size = 1543, normalized size = 2.42 \begin {gather*} \frac {c^2 \sqrt {c+a^2 c x^2} \left (35678 a x \sqrt {1+a^2 x^2}+24602 a^3 x^3 \sqrt {1+a^2 x^2}-4070 a^5 x^5 \sqrt {1+a^2 x^2}+7006 a^7 x^7 \sqrt {1+a^2 x^2}+21002 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)-49890 a^2 x^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)-109026 a^4 x^4 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)-38134 a^6 x^6 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+1273965 a x \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+2168775 a^3 x^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+1080135 a^5 x^5 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+185325 a^7 x^7 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+202902 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+439768 a^2 x^2 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+263172 a^4 x^4 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+18648 a^6 x^6 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))-7658 a^8 x^8 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))-51310 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-164920 a^2 x^2 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-186900 a^4 x^4 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-84280 a^6 x^6 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-10990 a^8 x^8 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))+3150 \text {ArcTan}(a x) \cos (7 \text {ArcTan}(a x))+12600 a^2 x^2 \text {ArcTan}(a x) \cos (7 \text {ArcTan}(a x))+18900 a^4 x^4 \text {ArcTan}(a x) \cos (7 \text {ArcTan}(a x))+12600 a^6 x^6 \text {ArcTan}(a x) \cos (7 \text {ArcTan}(a x))+3150 a^8 x^8 \text {ArcTan}(a x) \cos (7 \text {ArcTan}(a x))-100800 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-100800 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )+100800 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+100800 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-100800 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-100800 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )+100800 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+203264 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-100800 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-203264 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+100800 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+100800 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-201600 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+201600 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )+201600 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )-201600 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )+17622 \sin (3 \text {ArcTan}(a x))+11352 a^2 x^2 \sin (3 \text {ArcTan}(a x))-17916 a^4 x^4 \sin (3 \text {ArcTan}(a x))+600 a^6 x^6 \sin (3 \text {ArcTan}(a x))+12246 a^8 x^8 \sin (3 \text {ArcTan}(a x))-490455 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-1484700 a^2 x^2 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-1592010 a^4 x^4 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-691740 a^6 x^6 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-93975 a^8 x^8 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-15618 \sin (5 \text {ArcTan}(a x))-39176 a^2 x^2 \sin (5 \text {ArcTan}(a x))-23820 a^4 x^4 \sin (5 \text {ArcTan}(a x))+7416 a^6 x^6 \sin (5 \text {ArcTan}(a x))+7678 a^8 x^8 \sin (5 \text {ArcTan}(a x))+61845 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+227220 a^2 x^2 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+310590 a^4 x^4 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+186900 a^6 x^6 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+41685 a^8 x^8 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+2438 \sin (7 \text {ArcTan}(a x))+9752 a^2 x^2 \sin (7 \text {ArcTan}(a x))+14628 a^4 x^4 \sin (7 \text {ArcTan}(a x))+9752 a^6 x^6 \sin (7 \text {ArcTan}(a x))+2438 a^8 x^8 \sin (7 \text {ArcTan}(a x))-1575 \text {ArcTan}(a x)^2 \sin (7 \text {ArcTan}(a x))-6300 a^2 x^2 \text {ArcTan}(a x)^2 \sin (7 \text {ArcTan}(a x))-9450 a^4 x^4 \text {ArcTan}(a x)^2 \sin (7 \text {ArcTan}(a x))-6300 a^6 x^6 \text {ArcTan}(a x)^2 \sin (7 \text {ArcTan}(a x))-1575 a^8 x^8 \text {ArcTan}(a x)^2 \sin (7 \text {ArcTan}(a x))\right )}{2580480 a^3 \sqrt {1+a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.76, size = 376, normalized size = 0.59
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right )^{2} a^{7} x^{7}-1440 \arctan \left (a x \right ) a^{6} x^{6}+14280 \arctan \left (a x \right )^{2} a^{5} x^{5}+240 a^{5} x^{5}-3984 \arctan \left (a x \right ) a^{4} x^{4}+12390 \arctan \left (a x \right )^{2} a^{3} x^{3}+696 a^{3} x^{3}-2948 \arctan \left (a x \right ) a^{2} x^{2}+1575 \arctan \left (a x \right )^{2} a x +430 a x +2746 \arctan \left (a x \right )\right )}{40320 a^{3}}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1575 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6352 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40320 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(376\) |
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 x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, 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 = -0.00 \begin {gather*} \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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