Optimal. Leaf size=655 \[ \frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}-\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}+\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}+\frac{1}{12} a^2 c^2 x \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{7}{4} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{4 a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{4} a^2 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{1}{6} a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
[Out]
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Rubi [A] time = 1.4075, antiderivative size = 655, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4950, 4944, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206, 195} \[ \frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}-\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}+\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{a^2 c x^2+c}}+\frac{1}{12} a^2 c^2 x \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{7}{4} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{4 a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{4} a^2 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac{1}{6} a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4944
Rule 4958
Rule 4954
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4880
Rule 217
Rule 206
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{x^2} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^2} \, dx+\left (a^2 c\right ) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx+\frac{1}{6} \left (a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx+\frac{1}{4} \left (3 a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^3 \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{12} \left (a^2 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{4} \left (3 a^2 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{12} \left (a^2 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{1}{4} \left (3 a^2 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 )+\left (a^2 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 (3 a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (a^2 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}}+\frac{\left (a^2 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}}\\ &=\frac{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (a 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}}+\frac{\left (a 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 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}}\\ &=\frac{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{c+a^2 c x^2}}-\frac{4 a 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{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{2 i a 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 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 (3 a 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 )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a 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 )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (a 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 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 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 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{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{c+a^2 c x^2}}-\frac{4 a 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{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{2 i a 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 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 (3 i a 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 )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 i a 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 )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (i a 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 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 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 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{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{c+a^2 c x^2}}-\frac{4 a 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{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{2 i a 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 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 (3 a 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 )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a 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 )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (a 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 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 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 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{1}{12} a^2 c^2 x \sqrt{c+a^2 c x^2}-\frac{7}{4} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 \sqrt{c+a^2 c x^2}}-\frac{4 a 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{11}{6} a c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{2 i a 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 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{15 a c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{15 a c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.72348, size = 626, normalized size = 0.96 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (192 i a x \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+360 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-360 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-192 i a x \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-360 a x \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+360 a x \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+2 a^4 x^4 \sqrt{a^2 x^2+1}+2 a^2 x^2 \sqrt{a^2 x^2+1}+21 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+117 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-190 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-96 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+176 a x \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+2 a^5 x^5 \sin \left (3 \tan ^{-1}(a x)\right )-3 a^5 x^5 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+4 a^3 x^3 \sin \left (3 \tan ^{-1}(a x)\right )-6 a^3 x^3 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+6 a^5 x^5 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )+12 a^3 x^3 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-168 i a x \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+192 a x \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )+96 a x \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-96 a x \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-192 a x \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-3 a x \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+2 a x \sin \left (3 \tan ^{-1}(a x)\right )+6 a x \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )\right )}{96 x \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.426, size = 399, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 6\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-4\,\arctan \left ( ax \right ){x}^{3}{a}^{3}+27\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+2\,{a}^{2}{x}^{2}-46\,\arctan \left ( ax \right ) xa-24\, \left ( \arctan \left ( ax \right ) \right ) ^{2} \right ) }{24\,x}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{{\frac{i}{24}}a{c}^{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 45\,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 ) -45\,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 ) +48\,i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +90\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -90\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +90\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -90\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +48\,{\it dilog} \left ({\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +48\,{\it dilog} \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -88\,\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^{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] 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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