Optimal. Leaf size=531 \[ -\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{360 a^3}+\frac{1}{60} c x^3 \sqrt{a^2 c x^2+c}+\frac{c x \sqrt{a^2 c x^2+c}}{36 a^2}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{15} a c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{19 c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3} \]
[Out]
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Rubi [A] time = 3.18631, antiderivative size = 531, normalized size of antiderivative = 1., number of steps used = 92, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ -\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{360 a^3}+\frac{1}{60} c x^3 \sqrt{a^2 c x^2+c}+\frac{c x \sqrt{a^2 c x^2+c}}{36 a^2}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{15} a c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{19 c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4952
Rule 4930
Rule 217
Rule 206
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 321
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^4 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^2}+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c^2 \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a}+2 \left (\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} \left (a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\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)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (a^3 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^3}-\frac{1}{15} a 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^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{8} \left (5 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c^2 \int \frac{1}{\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)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} c^2 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}+\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a}+\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{4 a}\right )+\frac{1}{15} \left (4 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{12} \left (5 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{15} \left (a^2 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^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{20} c^2 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{45} \left (4 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{36} \left (5 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{\left (5 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{16 a^2}+\frac{c^2 \operatorname{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^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{45 a}-\frac{\left (5 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{18 a}-\frac{\left (5 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{8 a}-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \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}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{12 a^2}-\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{4 a^2}+\frac{\left (3 c^2 \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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{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^3 \sqrt{c+a^2 c x^2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3}+\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^2}+\frac{\left (2 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{45 a^2}+\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{72 a^2}+\frac{\left (8 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{45 a^2}+\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{18 a^2}+\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}+2 \left (\frac{c x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c^2 \operatorname{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{c^2 \operatorname{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 (3 c^2\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt{c+a^2 c x^2}}\right )+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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 (5 c^2 \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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{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^3 \sqrt{c+a^2 c x^2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3}-\frac{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^3 \sqrt{c+a^2 c x^2}}+\frac{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^3 \sqrt{c+a^2 c x^2}}+\frac{c^2 \operatorname{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^2\right ) \operatorname{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^2\right ) \operatorname{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^2\right ) \operatorname{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^2\right ) \operatorname{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^2\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt{c+a^2 c x^2}}+2 \left (\frac{c x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a^3}-\frac{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^3 \sqrt{c+a^2 c x^2}}+\frac{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^3 \sqrt{c+a^2 c x^2}}+2 \left (\frac{c x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}+\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 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 )}{4 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 \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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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}}\right )+\frac{\left (5 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a^3}-\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{c^2 \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^2 \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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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}}+2 \left (\frac{c x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}+\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 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 )}{4 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 \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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a^3}-\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{c^2 \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^2 \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}}+2 \left (\frac{c x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}+\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 c^2 \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^2 \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}}\right )+\frac{\left (5 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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^2 \sqrt{1+a^2 x^2}\right ) \operatorname{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{5 c x \sqrt{c+a^2 c x^2}}{36 a^2}+\frac{1}{60} c x^3 \sqrt{c+a^2 c x^2}-\frac{749 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac{41 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac{1}{15} a c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{13 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac{5}{24} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{799 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{360 a^3}-\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{13 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 )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{13 c^2 \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{13 c^2 \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 x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{13 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{7 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}+\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 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 )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{3 c^2 \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^2 \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}}\right )\\ \end{align*}
Mathematica [A] time = 3.46325, size = 527, normalized size = 0.99 \[ \frac{c \sqrt{a^2 c x^2+c} \left (960 \left (-3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+3 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-3 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+3 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+32 \left (45 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-45 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-45 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+45 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+19 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-45 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+\left (a^2 x^2+1\right )^3 \left (-\frac{56 a x}{\sqrt{a^2 x^2+1}}+15 \tan ^{-1}(a x)^2 \left (\frac{78 a x}{\sqrt{a^2 x^2+1}}-47 \sin \left (3 \tan ^{-1}(a x)\right )+3 \sin \left (5 \tan ^{-1}(a x)\right )\right )+\tan ^{-1}(a x) \left (\frac{12}{\sqrt{a^2 x^2+1}}+110 \cos \left (3 \tan ^{-1}(a x)\right )-90 \cos \left (5 \tan ^{-1}(a x)\right )\right )-108 \sin \left (3 \tan ^{-1}(a x)\right )-52 \sin \left (5 \tan ^{-1}(a x)\right )\right )+120 \left (a^2 x^2+1\right )^{3/2} \left (-3 \tan ^{-1}(a x)^2 \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )-7 a x\right )+2 \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )+a x\right )+\tan ^{-1}(a x) \left (6 \sqrt{a^2 x^2+1} \cos \left (3 \tan ^{-1}(a x)\right )+2\right )\right )\right )}{11520 a^3 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.487, size = 338, normalized size = 0.6 \begin{align*}{\frac{c \left ( 120\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}-48\,\arctan \left ( ax \right ){x}^{4}{a}^{4}+210\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+12\,{a}^{3}{x}^{3}-76\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+45\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+20\,ax+62\,\arctan \left ( ax \right ) \right ) }{720\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{\frac{i}{720}}c}{{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 45\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -45\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +90\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -90\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +90\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -90\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -164\,\arctan \left ({\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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 )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{2}{\left (a x \right )}\, dx \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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