Optimal. Leaf size=675 \[ \frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c^2 \sqrt{a^2 c x^2+c}}{3 x}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{26 a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3} \]
[Out]
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Rubi [A] time = 2.30888, antiderivative size = 675, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 16, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4950, 4944, 4946, 4962, 264, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206} \[ \frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c^2 \sqrt{a^2 c x^2+c}}{3 x}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{26 a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4944
Rule 4946
Rule 4962
Rule 264
Rule 4958
Rule 4954
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4880
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{x^4} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^4} \, dx+\left (a^2 c\right ) \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^2} \, dx\\ &=c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx\right )+\left (a^4 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} \left (2 a c^2\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+\frac{1}{2} \left (a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )\\ &=-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} \left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (2 a^2 c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\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 )+\frac{\left (a^4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\left (2 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{2 a^2 c^2 \sqrt{c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )-\frac{1}{3} \left (a^2 c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^3 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{\left (a^3 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a^2 c^2 \sqrt{c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^3 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{4 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a^2 c^2 \sqrt{c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{2 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (i a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (i a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{4 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (2 i a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 i a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a^2 c^2 \sqrt{c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{2 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{\left (a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{4 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (2 a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 c^3 \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 )}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a^2 c^2 \sqrt{c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{2 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}+a^3 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}}-\frac{4 a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{2 a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{2 a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 4.56544, size = 644, normalized size = 0.95 \[ -\frac{c^3 \sqrt{a^2 x^2+1} \left (-52 i a^3 x^3 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-60 i a^3 x^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+60 i a^3 x^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+52 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+60 a^3 x^3 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-60 a^3 x^3 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+2 \left (a^2 x^2+1\right )^{3/2}-6 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+12 i a^3 x^3 \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+12 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+24 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+4 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2-12 a^3 x^3 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-51 a^3 x^3 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+51 a^3 x^3 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+2 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-2 \left (a^2 x^2+1\right )^{3/2} \cos \left (2 \tan ^{-1}(a x)\right )+\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )-\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )-3 a x \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )+3 a x \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )\right )}{12 x^3 \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.556, size = 401, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-6\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-14\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-2\,{a}^{2}{x}^{2}-2\,\arctan \left ( ax \right ) xa-2\, \left ( \arctan \left ( ax \right ) \right ) ^{2} \right ) }{6\,{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{{\frac{i}{6}}{a}^{3}{c}^{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 15\,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 ) -15\,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 ) +26\,i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +30\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -30\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +30\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -30\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +26\,{\it dilog} \left ({\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +26\,{\it dilog} \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -12\,\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 (\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^{4}}, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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