Optimal. Leaf size=567 \[ -\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{4 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 1.65459, antiderivative size = 567, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4950, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886} \[ -\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{4 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4962
Rule 4944
Rule 266
Rule 63
Rule 208
Rule 4958
Rule 4956
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4930
Rule 4890
Rule 4886
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^3} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x} \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^2}{x^3 \sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\left (a c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} \left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx-\left (2 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+2 \frac{\left (a^2 c^2 \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{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\left (a^2 c^2\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{\left (a^2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+2 \frac{\left (a^2 c^2 \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 (2 a^3 c^2 \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{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\frac{4 i a^2 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 )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{1}{2} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )-\frac{\left (a^2 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{2 a^2 c^2 \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{\left (2 a^2 c^2 \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 a^2 c^2 \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}}\right )\\ &=-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\frac{4 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{a^2 c^2 \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 a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+c \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )+2 \left (-\frac{2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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{\left (2 i a^2 c^2 \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 a^2 c^2 \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}}\right )+\frac{\left (a^2 c^2 \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 (a^2 c^2 \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{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\frac{4 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{a^2 c^2 \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}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )-\frac{i a^2 c^2 \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{i a^2 c^2 \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 a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (i a^2 c^2 \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 (i a^2 c^2 \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}}+2 \left (-\frac{2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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{\left (2 a^2 c^2 \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 a^2 c^2 \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}}\right )\\ &=-\frac{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\frac{4 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{a^2 c^2 \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}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )-\frac{i a^2 c^2 \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{i a^2 c^2 \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 a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^2 \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 (a^2 c^2 \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{a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x^2}+\frac{4 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{a^2 c^2 \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}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )-\frac{i a^2 c^2 \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{i a^2 c^2 \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 a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^2 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 )}{\sqrt{c+a^2 c x^2}}+\frac{a^2 c^2 \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{a^2 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 2.9497, size = 455, normalized size = 0.8 \[ \frac{a^2 c \sqrt{a^2 c x^2+c} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \left (24 i \tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-24 i \tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-16 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+16 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-24 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+24 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )-4 \tan ^{-1}(a x)-4 \tan ^{-1}(a x) \cot ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+4 a x \tan ^{-1}(a x)^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-16 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )+16 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )+8 \log \left (\tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\tan ^{-1}(a x)^2 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+\tan ^{-1}(a x)^2 \csc \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{8 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.385, size = 412, normalized size = 0.7 \begin{align*}{\frac{c\arctan \left ( ax \right ) \left ( 2\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-2\,ax-\arctan \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{a}^{2}c}{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -4\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +4\,\arctan \left ( ax \right ) \ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +4\,i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -4\,i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,\ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it polylog} \left ( 3,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -2\,\ln \left ({\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}}-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^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{2}}{x^{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 \frac{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{2}{\left (a x \right )}}{x^{3}}\, dx \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{3}{2}} \arctan \left (a x\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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