Optimal. Leaf size=726 \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{2} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) \]
[Out]
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Rubi [A] time = 1.14392, antiderivative size = 726, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206} \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{2} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4958
Rule 4956
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4930
Rule 4890
Rule 4888
Rule 4181
Rule 4880
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-(a c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+c^2 \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^2\right ) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{1}{2} \left (a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\left (a c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\left (3 a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\left (a c^2\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 (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (a c^2 \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 (3 a c^2 \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}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{2 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )-\frac{\left (c^2 \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 (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 c^2 \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}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{7 i c^2 \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}}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{2 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (c^2 \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 (c^2 \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 (6 c^2 \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 (6 c^2 \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}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{7 i c^2 \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}}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{2 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{7 i c^2 \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{7 i c^2 \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{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (i c^2 \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 c^2 \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 (6 i c^2 \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 (6 i c^2 \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 (6 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{7 i c^2 \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}}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{2 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{7 i c^2 \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{7 i c^2 \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{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (c^2 \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 (c^2 \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 (6 c^2 \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 (6 c^2 \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}}\\ &=c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{1}{2} a c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{7 i c^2 \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}}+c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{2 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{7 i c^2 \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{7 i c^2 \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{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{7 c^2 \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{7 c^2 \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{6 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.3687, size = 555, normalized size = 0.76 \[ \frac{c \sqrt{a^2 c x^2+c} \left (72 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \tan ^{-1}(a x)}\right )+72 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-168 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+168 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+144 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \tan ^{-1}(a x)}\right )-144 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+168 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-168 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-144 i \text{PolyLog}\left (4,e^{-i \tan ^{-1}(a x)}\right )-144 i \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )+\frac{8 a^4 x^4 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{12 a^3 x^3 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{40 a^2 x^2 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{24 a^2 x^2 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{12 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{32 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{24 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-24 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+6 i \tan ^{-1}(a x)^4+24 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+24 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )-72 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+72 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-24 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-3 i \pi ^4\right )}{24 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.52, size = 511, normalized size = 0.7 \begin{align*} \text{result too large to display} \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 )^{3}}{x}, 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}^{3}{\left (a x \right )}}{x}\, 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 )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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