Optimal. Leaf size=476 \[ \frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{140 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{5/2}}{105 a^4 c}-\frac{17 \left (a^2 c x^2+c\right )^{3/2}}{1260 a^4}-\frac{17 c \sqrt{a^2 c x^2+c}}{280 a^4}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{21} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4} \]
[Out]
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Rubi [A] time = 4.07276, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 75, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4950, 4952, 261, 4890, 4886, 4930, 266, 43} \[ \frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{140 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{5/2}}{105 a^4 c}-\frac{17 \left (a^2 c x^2+c\right )^{3/2}}{1260 a^4}-\frac{17 c \sqrt{a^2 c x^2+c}}{280 a^4}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{21} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4952
Rule 261
Rule 4890
Rule 4886
Rule 4930
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{\left (2 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a}+2 \left (\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{5} \left (4 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{5} \left (2 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\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)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (2 a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}-\frac{1}{21} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{35} \left (24 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}+\frac{\left (4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^3}+\frac{c^2 \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}+2 \left (-\frac{c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{10} c^2 \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{10 a}+\frac{\left (8 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a}\right )+\frac{1}{21} \left (5 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (12 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{21} \left (a^2 c^2\right ) \int \frac{x^5}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c \sqrt{c+a^2 c x^2}}{3 a^4}-\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{84} \left (5 c^2\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{35} \left (3 c^2\right ) \int \frac{x^3}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{20} c^2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}-\frac{\left (4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (16 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^3}-\frac{\left (3 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^2}-\frac{\left (4 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}\right )-\frac{\left (16 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}-\frac{\left (5 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{28 a}-\frac{\left (9 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}-\frac{\left (16 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}+\frac{1}{42} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{c \sqrt{c+a^2 c x^2}}{3 a^4}-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{10 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}+\frac{5 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}-\frac{5 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}-\frac{1}{168} \left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{1}{70} \left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+a^2 c x}} \, dx,x,x^2\right )+\frac{\left (5 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^3}+\frac{\left (9 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^3}+\frac{\left (8 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (32 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (5 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^2}+\frac{\left (9 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^2}+\frac{\left (8 c^2\right ) \int \frac{x}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+\frac{1}{42} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{c+a^2 c x}}-\frac{2 \sqrt{c+a^2 c x}}{a^4 c}+\frac{\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+2 \left (-\frac{5 c \sqrt{c+a^2 c x^2}}{12 a^4}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{20} c^2 \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 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{20 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{15 a^3 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{139 c \sqrt{c+a^2 c x^2}}{168 a^4}-\frac{2 \left (c+a^2 c x^2\right )^{3/2}}{63 a^4}+\frac{\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{10 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}+\frac{5 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}-\frac{5 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 )}{3 a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{31 c \sqrt{c+a^2 c x^2}}{60 a^4}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{89 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 )}{30 a^4 \sqrt{c+a^2 c x^2}}-\frac{89 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 )}{60 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 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 )}{60 a^4 \sqrt{c+a^2 c x^2}}\right )-\frac{1}{168} \left (5 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{1}{70} \left (3 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 (5 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{56 a^3 \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}{70 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)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (32 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{817 c \sqrt{c+a^2 c x^2}}{840 a^4}-\frac{101 \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac{\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac{1}{21} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{7} a^2 c x^6 \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}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{420 a^4 \sqrt{c+a^2 c x^2}}+\frac{2543 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 )}{840 a^4 \sqrt{c+a^2 c x^2}}-\frac{2543 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 )}{840 a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{31 c \sqrt{c+a^2 c x^2}}{60 a^4}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{89 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 )}{30 a^4 \sqrt{c+a^2 c x^2}}-\frac{89 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 )}{60 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 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 )}{60 a^4 \sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 4.56689, size = 797, normalized size = 1.67 \[ \frac{c \left (a^2 x^2+1\right )^2 \sqrt{a^2 c x^2+c} \left (\left (a^2 x^2+1\right ) \left (-5376 \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2+6720 \cos \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2+10944 \tan ^{-1}(a x)^2-6489 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-2163 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-309 \cos \left (7 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac{10815 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+6489 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+2163 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+309 \cos \left (7 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac{10815 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-1266 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+360 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-618 \sin \left (6 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+6262 \cos \left (2 \tan ^{-1}(a x)\right )+2764 \cos \left (4 \tan ^{-1}(a x)\right )+618 \cos \left (6 \tan ^{-1}(a x)\right )-\frac{19776 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{7/2}}+\frac{19776 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{7/2}}+4116\right )-168 \left (160 \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac{110 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac{110 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+4 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-22 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+72 \cos \left (2 \tan ^{-1}(a x)\right )+22 \cos \left (4 \tan ^{-1}(a x)\right )-\frac{176 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+\frac{176 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+50\right )\right )}{161280 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.909, size = 271, normalized size = 0.6 \begin{align*}{\frac{c \left ( 360\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{6}{a}^{6}-120\,\arctan \left ( ax \right ){x}^{5}{a}^{5}+576\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}+24\,{a}^{4}{x}^{4}-138\,\arctan \left ( ax \right ){x}^{3}{a}^{3}+72\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+14\,{a}^{2}{x}^{2}+135\,\arctan \left ( ax \right ) xa-144\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-163 \right ) }{2520\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{17\,c}{280\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\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^{5} + c x^{3}\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(-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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