Optimal. Leaf size=760 \[ \frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{4 a}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac{1}{4} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) \]
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Rubi [A] time = 0.524236, antiderivative size = 760, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886, 4878} \[ \frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{4 a}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac{1}{4} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4880
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4886
Rule 4878
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{2} c \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{4} c^2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{4} \left (9 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}+\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c \sqrt{c+a^2 c x^2}}{4 a}+\frac{1}{4} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{3 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{a \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{2 a \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [B] time = 12.9469, size = 2105, normalized size = 2.77 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.434, size = 466, normalized size = 0.6 \begin{align*}{\frac{c \left ( 2\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}-2\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+5\, \left ( \arctan \left ( ax \right ) \right ) ^{3}ax+2\,\arctan \left ( ax \right ) xa-11\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{8\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{c}{8\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 3\, \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +9\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +20\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +18\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -20\,\arctan \left ( ax \right ) \ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -18\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +18\,i{\it polylog} \left ( 4,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -18\,i{\it polylog} \left ( 4,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -20\,i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +20\,i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\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 ({\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{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 \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{3}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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