Optimal. Leaf size=652 \[ \frac {c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}-\frac {163 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{840 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)+\frac {9 c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2-\frac {51 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {23 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {51 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {51 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {51 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}+\frac {51 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 4.92, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps
used = 200, 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 {c x^2 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}}{35 a^2}+\frac {c x^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{60 a^2}+\frac {1}{7} a^2 c x^6 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}-\frac {1}{14} a c x^5 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}+\frac {8}{35} c x^4 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}+\frac {1}{35} c x^4 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}-\frac {23 c x^3 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{280 a}-\frac {c x^3 \sqrt {a^2 c x^2+c}}{140 a}+\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}+\frac {51 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {2 c \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}}{35 a^4}-\frac {163 c \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{840 a^4}+\frac {23 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^4}+\frac {9 c x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{112 a^3}+\frac {c x \sqrt {a^2 c x^2+c}}{420 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^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=c^2 \int \frac {x^3 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^7 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^2}+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {c^2 \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a}+2 \left (\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{5} \left (4 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (3 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (6 a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (3 a^3 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{35} \left (24 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^3}+\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}+\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{10} \left (3 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^2\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (9 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a}+\frac {\left (4 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{5 a}\right )+\frac {1}{14} \left (5 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (18 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{7} \left (a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^4}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{35} \left (4 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{28} \left (5 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (9 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}-\frac {\left (16 c^2\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+2 \left (\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {\left (9 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^3}-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^3}-\frac {\left (8 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^3}-\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^2}-\frac {\left (9 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {\left (4 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^2}-\frac {c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}\right )-\frac {\left (15 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{56 a}-\frac {\left (27 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{70 a}-\frac {\left (24 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {1}{35} \left (a c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {\left (15 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{112 a^3}+\frac {\left (27 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{140 a^3}+\frac {\left (12 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}-\frac {c^2 \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^3}+\frac {\left (48 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (8 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^2}+\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{42 a^2}+\frac {\left (6 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {\left (15 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^2}+\frac {\left (27 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^2}+\frac {\left (24 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {\left (3 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{140 a}+\frac {\left (4 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{105 a}+\frac {\left (5 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{84 a}+\frac {\left (3 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (2 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^3}+\frac {\left (9 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}+\frac {\left (4 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{5 a^3}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{40 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{5 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{5 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{280 a^3}-\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^3}-\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{168 a^3}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}-\frac {\left (8 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{105 a^3}-\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{42 a^3}-\frac {\left (6 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}-\frac {\left (15 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^3}-\frac {\left (27 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}-\frac {\left (24 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}-\frac {\left (c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (c^2 \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^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{20 a^3}+\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{5 a^3}+\frac {\left (9 c^2\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 )}{20 a^3}+\frac {\left (4 c^2\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 )}{5 a^3}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{40 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (8 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (4 c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{112 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (27 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{140 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (12 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (48 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2\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 )}{280 a^3}-\frac {\left (2 c^2\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 )}{105 a^3}-\frac {\left (5 c^2\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 )}{168 a^3}-\frac {\left (3 c^2\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 )}{70 a^3}-\frac {\left (8 c^2\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 )}{105 a^3}-\frac {\left (5 c^2\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 )}{42 a^3}-\frac {\left (6 c^2\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^3}-\frac {\left (15 c^2\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 )}{56 a^3}-\frac {\left (27 c^2\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 )}{70 a^3}-\frac {\left (24 c^2\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^3}-\frac {\left (i c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (i c^2 \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^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 i c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 i c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{112 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (27 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{140 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (12 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (48 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}+\frac {\left (9 c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (16 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {337 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 i c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 i c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (16 i c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (16 i c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (15 c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (27 c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (27 c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (24 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (24 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (c^2 \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^4 \sqrt {c+a^2 c x^2}}-\frac {\left (96 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (96 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^2 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^2 \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^4 \sqrt {c+a^2 c x^2}}\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {337 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+\frac {5 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (27 i c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (27 i c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (24 i c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (24 i c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (96 i c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (96 i c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \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 )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (16 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c^2 \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 )}{5 a^4 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {337 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+\frac {5 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (15 c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \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 )}{56 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (27 c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (27 c^2 \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 )}{70 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (24 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (24 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}-\frac {\left (96 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (96 c^2 \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 )}{35 a^4 \sqrt {c+a^2 c x^2}}\\ &=\frac {43 c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}+\frac {2273 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac {11 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {337 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}+\frac {2543 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {29 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {3 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}-\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 7.72, size = 1296, normalized size = 1.99 \begin {gather*} \frac {c \left (\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (11 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+11 \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 )-11 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-11 \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 )+11 \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 )+11 \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 )-11 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-20 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+11 \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 )+20 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-11 \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 )-11 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+22 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-22 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-22 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+22 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )\right )}{40 \sqrt {1+a^2 x^2}}-\frac {1}{960} \left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (150 \text {ArcTan}(a x)-32 \text {ArcTan}(a x)^3+8 \text {ArcTan}(a x) \left (27+20 \text {ArcTan}(a x)^2\right ) \cos (2 \text {ArcTan}(a x))+66 \text {ArcTan}(a x) \cos (4 \text {ArcTan}(a x))+12 \sin (2 \text {ArcTan}(a x))+6 \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))+6 \sin (4 \text {ArcTan}(a x))-33 \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))\right )\right )}{a^4}+\frac {c \left (-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (309 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+309 \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 )-309 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-309 \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 )+309 \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 )+309 \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 )-309 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-518 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+309 \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 )+518 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-309 \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 )-309 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+618 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-618 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-618 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+618 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )\right )}{1680 \sqrt {1+a^2 x^2}}-\frac {\left (1+a^2 x^2\right )^3 \sqrt {c \left (1+a^2 x^2\right )} \left (-4116 \text {ArcTan}(a x)-3648 \text {ArcTan}(a x)^3+2 \text {ArcTan}(a x) \left (-3131+896 \text {ArcTan}(a x)^2\right ) \cos (2 \text {ArcTan}(a x))-4 \text {ArcTan}(a x) \left (691+560 \text {ArcTan}(a x)^2\right ) \cos (4 \text {ArcTan}(a x))-618 \text {ArcTan}(a x) \cos (6 \text {ArcTan}(a x))-404 \sin (2 \text {ArcTan}(a x))+633 \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))-352 \sin (4 \text {ArcTan}(a x))-180 \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))-100 \sin (6 \text {ArcTan}(a x))+309 \text {ArcTan}(a x)^2 \sin (6 \text {ArcTan}(a x))\right )}{53760}\right )}{a^4} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
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Maple [A]
time = 7.88, size = 469, normalized size = 0.72
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 \arctan \left (a x \right )^{3} a^{6} x^{6}-120 \arctan \left (a x \right )^{2} a^{5} x^{5}+384 \arctan \left (a x \right )^{3} a^{4} x^{4}+48 \arctan \left (a x \right ) a^{4} x^{4}-138 \arctan \left (a x \right )^{2} a^{3} x^{3}+48 \arctan \left (a x \right )^{3} a^{2} x^{2}-12 a^{3} x^{3}+28 \arctan \left (a x \right ) a^{2} x^{2}+135 \arctan \left (a x \right )^{2} a x -96 \arctan \left (a x \right )^{3}+4 a x -326 \arctan \left (a x \right )\right )}{1680 a^{4}}+\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {23 i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(469\) |
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^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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