Optimal. Leaf size=523 \[ \frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}+\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c}}{20 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{20 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^2}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^4}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{2 a^4} \]
[Out]
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Rubi [A] time = 2.4372, antiderivative size = 523, normalized size of antiderivative = 1., number of steps used = 71, 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} \[ \frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}+\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c}}{20 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{20 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^2}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^4}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{2 a^4} \]
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^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx &=c \int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac{x^5 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{1}{5} (4 c) \int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{(2 c) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{c \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a}-\frac{1}{5} (3 a c) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{10} (3 c) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^3}+\frac{(2 c) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a^3}+\frac{(8 c) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{c \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}+\frac{(9 c) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}+\frac{(4 c) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a}\\ &=\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{(9 c) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^3}-\frac{(2 c) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{a^3}-\frac{(8 c) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{c \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^2}-\frac{(9 c) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^2}-\frac{(4 c) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^2}-\frac{c \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{10 a}+\frac{\left (c \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 \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{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}+\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}+\frac{(9 c) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}+\frac{(4 c) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{c \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^3}+\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{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 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c \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 \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 \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{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{5 i c \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{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^4}+\frac{c \operatorname{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 \operatorname{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{(9 c) \operatorname{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{(4 c) \operatorname{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 \sqrt{1+a^2 x^2}\right ) \operatorname{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 \sqrt{1+a^2 x^2}\right ) \operatorname{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 (c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (c \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^4 \sqrt{c+a^2 c x^2}}-\frac{\left (8 c \sqrt{1+a^2 x^2}\right ) \operatorname{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 (4 c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c \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^4 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{11 i c \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{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}+\frac{5 i c \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 \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 (i c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (i c \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^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 i c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 i c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{11 i c \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{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}+\frac{11 i c \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{11 i c \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 \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 i c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 i c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 i c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 i c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (c \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^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c \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^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c \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^4 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{11 i c \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{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}+\frac{11 i c \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{11 i c \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{5 c \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 \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 (9 c \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c \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 )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c \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 )}{5 a^4 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{9 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{11 i c \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{2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}+\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}+\frac{11 i c \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{11 i c \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{11 c \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{11 c \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}}\\ \end{align*}
Mathematica [A] time = 1.16379, size = 262, normalized size = 0.5 \[ \frac{\sqrt{a^2 c x^2+c} \left (-\left (a^2 x^2+1\right )^2 \left (\frac{48 a x}{\left (a^2 x^2+1\right )^2}+\tan ^{-1}(a x)^2 \left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right )+32 \tan ^{-1}(a x)^3 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right )+6 \tan ^{-1}(a x) \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )\right )+\frac{48 \left (11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-11 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+11 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+10 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )}{\sqrt{a^2 x^2+1}}\right )}{960 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 4.634, size = 417, normalized size = 0.8 \begin{align*}{\frac{24\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}{a}^{4}-18\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+8\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}+12\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+15\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa-16\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,ax-54\,\arctan \left ( ax \right ) }{120\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{11}{120\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}-3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{11}{120\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}+6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{i}{{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \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 (\sqrt{a^{2} c x^{2} + c} x^{3} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}^{3}{\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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