Optimal. Leaf size=882 \[ -\frac {c \sqrt {c+a^2 c x^2}}{30 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)+\frac {31 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{240 a^3}-\frac {19 c x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {41 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {41 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {41 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 3.72, antiderivative size = 882, normalized size of antiderivative = 1.00, number of steps
used = 108, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used
= {5070, 5072, 5050, 5010, 5006, 5008, 4266, 2611, 6744, 2320, 6724, 267, 272, 45}
\begin {gather*} \frac {1}{6} a^2 c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^5-\frac {1}{10} a c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x^4+\frac {7}{24} c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^3+\frac {1}{20} c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) x^3-\frac {19 c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x^2}{120 a}+\frac {c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x}{16 a^2}+\frac {c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) x}{12 a^2}+\frac {i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {31 c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2}{240 a^3}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{60 a^3}+\frac {41 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}-\frac {41 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {a^2 c x^2+c}}+\frac {41 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 c x^2+c}}{30 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 267
Rule 272
Rule 2320
Rule 2611
Rule 4266
Rule 5006
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=c^2 \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c^2 \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+2 \left (\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{4} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (3 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{6} \left (5 a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{2} c^2 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^2 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {\left (9 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}\right )+\frac {1}{5} \left (2 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{5} \left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\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}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{20} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{15} \left (4 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {\left (4 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}-\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{12 a}-\frac {\left (15 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a}-\frac {1}{20} \left (a c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\left (9 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c^2 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\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 a^2 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 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^3 \sqrt {c+a^2 c x^2}}+\frac {3 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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 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 )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{24 a^2}+\frac {\left (8 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{6 a^2}+\frac {\left (15 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {\left (3 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{40 a}+\frac {\left (2 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {\left (5 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{24 a}-\frac {1}{40} \left (a c^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\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 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \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 a^2 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {5 c \sqrt {c+a^2 c x^2}}{12 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 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^3 \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 (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \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 (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {1}{40} \left (a c^2\right ) \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{40 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{24 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{6 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {7 c \sqrt {c+a^2 c x^2}}{15 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{60 a^3 \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 (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \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 (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {7 c \sqrt {c+a^2 c x^2}}{15 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=\frac {7 c \sqrt {c+a^2 c x^2}}{15 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {7 c \sqrt {c+a^2 c x^2}}{15 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (15 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {7 c \sqrt {c+a^2 c x^2}}{15 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 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 )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {799 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 )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {39 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {39 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c \sqrt {c+a^2 c x^2}}{4 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}-\frac {7 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^3 \sqrt {c+a^2 c x^2}}+\frac {7 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^3 \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^3 \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^3 \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^3 \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^3 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(4015\) vs. \(2(882)=1764\).
time = 18.19, size = 4015, normalized size = 4.55 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 3.95, size = 514, normalized size = 0.58
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (40 \arctan \left (a x \right )^{3} a^{5} x^{5}-24 \arctan \left (a x \right )^{2} a^{4} x^{4}+70 \arctan \left (a x \right )^{3} a^{3} x^{3}+12 \arctan \left (a x \right ) a^{3} x^{3}-38 \arctan \left (a x \right )^{2} a^{2} x^{2}+15 \arctan \left (a x \right )^{3} a x -4 a^{2} x^{2}+20 \arctan \left (a x \right ) a x +31 \arctan \left (a x \right )^{2}-12\right )}{240 a^{3}}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+45 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+82 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-82 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-82 i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+82 i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{240 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(514\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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