Optimal. Leaf size=870 \[ -\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3 c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{60 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{16 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{16 a \sqrt{a^2 c x^2+c}}+\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{15 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2}{16 a}+\frac{17}{60} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{17 \sqrt{a^2 c x^2+c} c^2}{60 a}+\frac{5}{24} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac{5 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c}{24 a}-\frac{\left (a^2 c x^2+c\right )^{3/2} c}{60 a}+\frac{1}{20} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3-\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{10 a} \]
[Out]
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Rubi [A] time = 0.792093, antiderivative size = 870, normalized size of antiderivative = 1., number of steps used = 23, 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} \[ -\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3 c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{60 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{16 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{16 a \sqrt{a^2 c x^2+c}}+\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{15 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2}{16 a}+\frac{17}{60} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{17 \sqrt{a^2 c x^2+c} c^2}{60 a}+\frac{5}{24} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac{5 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c}{24 a}-\frac{\left (a^2 c x^2+c\right )^{3/2} c}{60 a}+\frac{1}{20} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3-\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{10 a} \]
Antiderivative was successfully verified.
[In]
[Out]
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 )^{5/2} \tan ^{-1}(a x)^3 \, dx &=-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3+\frac{1}{5} c \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3+\frac{1}{20} \left (3 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac{1}{12} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3+\frac{1}{40} \left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{24} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{16} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (15 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3+\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{40 \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{24 \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{16 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{\left (15 c^3 \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 )}{16 a \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^3 \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 )}{16 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{\left (15 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (15 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}-\frac{\left (15 c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{\left (15 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{\left (15 i c^3 \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{17 c^2 \sqrt{c+a^2 c x^2}}{60 a}-\frac{c \left (c+a^2 c x^2\right )^{3/2}}{60 a}+\frac{17}{60} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{20} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{15 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{24 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{10 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \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 )}{60 a \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a \sqrt{c+a^2 c x^2}}+\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{259 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{120 a \sqrt{c+a^2 c x^2}}-\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{15 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}-\frac{15 i c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{15 i c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [B] time = 18.9039, size = 4281, normalized size = 4.92 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.278, size = 518, normalized size = 0.6 \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 ({\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 )^{3}, 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(-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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