Optimal. Leaf size=605 \[ -\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{a^2 c x^2+c}}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{a^2 c x^2+c}}+\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{29}{60} c^2 \sqrt{a^2 c x^2+c}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{30 \sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{29}{60} a c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{30} c \left (a^2 c x^2+c\right )^{3/2}+\frac{1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{1}{10} a c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2 \]
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Rubi [A] time = 1.263, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {4950, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886, 4878} \[ -\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{a^2 c x^2+c}}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{a^2 c x^2+c}}+\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{29}{60} c^2 \sqrt{a^2 c x^2+c}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{30 \sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{29}{60} a c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{30} c \left (a^2 c x^2+c\right )^{3/2}+\frac{1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{1}{10} a c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4958
Rule 4956
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4930
Rule 4890
Rule 4886
Rule 4878
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{x} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{1}{5} (2 a c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx+c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{1}{10} \left (3 a c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx-\frac{1}{3} \left (2 a c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+c^3 \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac{1}{20} \left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (a c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{20 \sqrt{c+a^2 c x^2}}-\frac{\left (a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \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 \sqrt{c+a^2 c x^2}}-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}-\frac{\left (2 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \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 \sqrt{c+a^2 c x^2}}-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}-\frac{\left (2 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \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 \sqrt{c+a^2 c x^2}}-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}-\frac{\left (2 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=\frac{29}{60} c^2 \sqrt{c+a^2 c x^2}+\frac{1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac{29}{60} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \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 \sqrt{c+a^2 c x^2}}-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{60 \sqrt{c+a^2 c x^2}}-\frac{2 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 7.09301, size = 889, normalized size = 1.47 \[ \sqrt{c \left (a^2 x^2+1\right )} \left (\frac{\left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\tan ^{-1}(a x)^2+\frac{2 i \left (\text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{2 \left (\tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )+i \left (\text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )\right )\right )}{\sqrt{a^2 x^2+1}}+\frac{2 \left (\text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt{a^2 x^2+1}}\right ) c^2+\frac{1}{6} \left (a^2 x^2+1\right ) \sqrt{c \left (a^2 x^2+1\right )} \left (4 \tan ^{-1}(a x)^2-\cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac{3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac{3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-2 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac{4 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac{4 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+2\right ) c^2-\frac{1}{960} \left (a^2 x^2+1\right )^2 \sqrt{c \left (a^2 x^2+1\right )} \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 ) c^2 \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.422, size = 404, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 12\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-6\,\arctan \left ( ax \right ){x}^{3}{a}^{3}+44\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+2\,{a}^{2}{x}^{2}-35\,\arctan \left ( ax \right ) xa+92\, \left ( \arctan \left ( ax \right ) \right ) ^{2}+31 \right ) }{60}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{{\frac{i}{60}}{c}^{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 60\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -60\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +149\,i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -149\,i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +120\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -120\,\arctan \left ( ax \right ){\it polylog} \left ( 2,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +120\,i{\it polylog} \left ( 3,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -120\,i{\it polylog} \left ( 3,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +149\,{\it dilog} \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -149\,{\it dilog} \left ( 1-{\frac{i \left ( 1+iax \right ) }{\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 (\frac{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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