Optimal. Leaf size=882 \[ \frac{1}{6} a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5-\frac{1}{10} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{7}{24} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{1}{20} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{19 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{120 a}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{16 a^2}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{12 a^2}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{240 a^3}-\frac{\left (a^2 c x^2+c\right )^{3/2}}{60 a^3}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{30 a^3} \]
[Out]
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Rubi [A] time = 5.46991, antiderivative size = 882, normalized size of antiderivative = 1., number of steps used = 108, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261, 266, 43} \[ \frac{1}{6} a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5-\frac{1}{10} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{7}{24} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{1}{20} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{19 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{120 a}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{16 a^2}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{12 a^2}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{240 a^3}-\frac{\left (a^2 c x^2+c\right )^{3/2}}{60 a^3}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{30 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4952
Rule 4930
Rule 4890
Rule 4886
Rule 4888
Rule 4181
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=c^2 \int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{c^2 \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a}+2 \left (\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{1}{4} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{4} \left (3 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )-\frac{1}{6} \left (5 a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} \left (a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{8} \left (5 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{2} c^2 \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}+\frac{c^2 \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a}+\frac{\left (9 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{8 a}\right )+\frac{1}{5} \left (2 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (5 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{5} \left (a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\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)^3}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{3 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{1}{20} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{15} \left (4 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{12} \left (5 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (5 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{16 a^2}-\frac{\left (4 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}-\frac{\left (5 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{12 a}-\frac{\left (15 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{16 a}-\frac{1}{20} \left (a c^2\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+2 \left (\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{4 a^2}-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}-\frac{\left (9 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{4 a^2}-\frac{c^2 \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{4 a}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2 \sqrt{c+a^2 c x^2}}\right )+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^2}+\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (5 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{24 a^2}+\frac{\left (8 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (5 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{6 a^2}+\frac{\left (15 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}+\frac{\left (3 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{40 a}+\frac{\left (2 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}+\frac{\left (5 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{24 a}-\frac{1}{40} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (5 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{16 a^2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{5 c \sqrt{c+a^2 c x^2}}{12 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{1}{40} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{c+a^2 c x}}+\frac{\sqrt{c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac{\left (3 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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 (3 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \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 (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \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 (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{40 a^2 \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)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{24 a^2 \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)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{6 a^2 \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)}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{7 c \sqrt{c+a^2 c x^2}}{15 a^3}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{13 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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 (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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}}\right )+\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{7 c \sqrt{c+a^2 c x^2}}{15 a^3}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{13 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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 (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\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 (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{7 c \sqrt{c+a^2 c x^2}}{15 a^3}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{13 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\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 (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{7 c \sqrt{c+a^2 c x^2}}{15 a^3}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{13 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\right )+\frac{\left (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{7 c \sqrt{c+a^2 c x^2}}{15 a^3}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac{1}{20} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac{1}{10} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{13 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}-\frac{799 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{39 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{39 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c \sqrt{c+a^2 c x^2}}{4 a^3}+\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{7 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [B] time = 18.2558, size = 4015, normalized size = 4.55 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.363, size = 514, normalized size = 0.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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