Optimal. Leaf size=578 \[ \frac {c^2 x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac {19}{63} a^2 c^2 x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {103 a c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac {5}{21} c^2 x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {205 c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2016 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^2 \sqrt {a^2 c x^2+c}}{4032 a^4}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}+\frac {1}{9} a^4 c^2 x^8 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac {\left (a^2 c x^2+c\right )^{7/2}}{252 a^4 c}-\frac {23 \left (a^2 c x^2+c\right )^{5/2}}{7560 a^4}-\frac {115 c \left (a^2 c x^2+c\right )^{3/2}}{18144 a^4}+\frac {47 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac {1}{36} a^3 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) \]
[Out]
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Rubi [A] time = 10.70, antiderivative size = 578, normalized size of antiderivative = 1.00, number of steps used = 203, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4950, 4952, 261, 4890, 4886, 4930, 266, 43} \[ \frac {115 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^2 \sqrt {a^2 c x^2+c}}{4032 a^4}+\frac {1}{9} a^4 c^2 x^8 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{36} a^3 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {19}{63} a^2 c^2 x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {103 a c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac {5}{21} c^2 x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {205 c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac {c^2 x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac {47 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2016 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{7/2}}{252 a^4 c}-\frac {23 \left (a^2 c x^2+c\right )^{5/2}}{7560 a^4}-\frac {115 c \left (a^2 c x^2+c\right )^{3/2}}{18144 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 261
Rule 266
Rule 4886
Rule 4890
Rule 4930
Rule 4950
Rule 4952
Rubi steps
\begin {align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^3 \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^9 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}+\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (2 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (2 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}-\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{5} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{5} \left (4 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (6 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{7} \left (2 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (2 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{9} \left (8 a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{9} \left (2 a^5 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}-\frac {c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{10} c^3 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\left (4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+\frac {\left (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {1}{21} \left (5 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{10} c^3 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (24 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {1}{21} \left (5 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{21} \left (16 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{36} \left (7 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{63} \left (16 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{36} \left (a^4 c^3\right ) \int \frac {x^7}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 \sqrt {c+a^2 c x^2}}{3 a^4}+\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}+\frac {41 a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}+\frac {31 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{20} c^3 \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{84} \left (5 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (3 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{105} \left (64 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {\left (4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (16 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (3 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {\left (4 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {\left (16 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{28 a}-\frac {\left (9 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {\left (16 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {1}{216} \left (35 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{189} \left (40 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{105} \left (32 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{42} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+2 \left (\frac {5 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {8 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{20} c^3 \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{84} \left (5 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (3 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {\left (4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (16 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (3 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {\left (4 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {\left (16 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{28 a}-\frac {\left (9 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {\left (16 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {1}{42} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )\right )-\frac {1}{216} \left (7 a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{189} \left (8 a^2 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{72} \left (a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2}}{12 a^4}-\frac {61 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}-\frac {3761 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac {41 a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {62 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {10 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 )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 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 )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 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 )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {1}{168} \left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{864} \left (35 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{70} \left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{20} c^3 \operatorname {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 {1}{189} \left (10 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{105} \left (8 c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^3}+\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}+\frac {\left (8 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (32 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^2}+\frac {\left (9 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^2}+\frac {\left (8 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {\left (128 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{315 a^2}+\frac {\left (35 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{288 a}+\frac {\left (10 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{63 a}+\frac {\left (8 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {\left (128 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{315 a}-\frac {1}{432} \left (7 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{189} \left (4 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{42} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+\frac {1}{72} \left (a^4 c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^6 \sqrt {c+a^2 c x}}+\frac {3 \sqrt {c+a^2 c x}}{a^6 c}-\frac {3 \left (c+a^2 c x\right )^{3/2}}{a^6 c^2}+\frac {\left (c+a^2 c x\right )^{5/2}}{a^6 c^3}\right ) \, dx,x,x^2\right )-\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}{20 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {5 c^2 \sqrt {c+a^2 c x^2}}{12 a^4}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {8 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{168} \left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{70} \left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{20} c^3 \operatorname {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 (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^3}+\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}+\frac {\left (8 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (32 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^2}+\frac {\left (9 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^2}+\frac {\left (8 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {1}{42} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\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}{20 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (16 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {713 c^2 \sqrt {c+a^2 c x^2}}{2520 a^4}+\frac {37 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{140 a^4}+\frac {\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac {127 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac {3761 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac {41 a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {58 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {11 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 )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )}{1728}+\frac {1}{189} \left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{168} \left (5 c^3\right ) \operatorname {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 {1}{105} \left (4 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{70} \left (3 c^3\right ) \operatorname {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 (35 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{576 a^3}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{63 a^3}-\frac {\left (4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}-\frac {\left (64 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{315 a^3}-\frac {\left (256 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{315 a^3}-\frac {\left (35 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{576 a^2}-\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{63 a^2}-\frac {\left (4 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (64 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{315 a^2}-\frac {1}{432} \left (7 a^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\frac {1}{189} \left (4 a^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+\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}{56 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{70 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (32 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {19 c^2 \sqrt {c+a^2 c x^2}}{840 a^4}+\frac {c \left (c+a^2 c x^2\right )^{3/2}}{630 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {8 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {89 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 )}{30 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {1}{168} \left (5 c^3\right ) \operatorname {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 {1}{70} \left (3 c^3\right ) \operatorname {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 (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{56 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{70 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (32 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {6299 c^2 \sqrt {c+a^2 c x^2}}{60480 a^4}+\frac {349 c \left (c+a^2 c x^2\right )^{3/2}}{11340 a^4}-\frac {167 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac {\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac {127 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac {3761 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac {41 a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {58 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1297 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 )}{420 a^4 \sqrt {c+a^2 c x^2}}+\frac {1297 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}-\frac {1297 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}+2 \left (\frac {103 c^2 \sqrt {c+a^2 c x^2}}{840 a^4}-\frac {59 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {8 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {103 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 )}{420 a^4 \sqrt {c+a^2 c x^2}}-\frac {103 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}+\frac {103 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (35 c^3\right ) \operatorname {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 )}{1728}+\frac {1}{189} \left (5 c^3\right ) \operatorname {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 {1}{105} \left (4 c^3\right ) \operatorname {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 (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{576 a^3 \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}{63 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (64 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{315 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (256 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{315 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {5519 c^2 \sqrt {c+a^2 c x^2}}{20160 a^4}+\frac {7921 c \left (c+a^2 c x^2\right )^{3/2}}{90720 a^4}-\frac {167 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac {\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac {127 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^3}-\frac {3761 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{30240 a}+\frac {41 a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {58 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^4}+\frac {29 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{315 a^2}+\frac {19}{105} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {5519 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 )}{10080 a^4 \sqrt {c+a^2 c x^2}}+\frac {5519 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 )}{20160 a^4 \sqrt {c+a^2 c x^2}}-\frac {5519 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 )}{20160 a^4 \sqrt {c+a^2 c x^2}}+2 \left (\frac {103 c^2 \sqrt {c+a^2 c x^2}}{840 a^4}-\frac {59 c \left (c+a^2 c x^2\right )^{3/2}}{1260 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {19 c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {8 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}-\frac {4 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}+\frac {1}{35} c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {103 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 )}{420 a^4 \sqrt {c+a^2 c x^2}}-\frac {103 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}+\frac {103 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 8.76, size = 1320, normalized size = 2.28 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.64, size = 309, normalized size = 0.53 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (20160 \arctan \left (a x \right )^{2} x^{8} a^{8}-5040 \arctan \left (a x \right ) x^{7} a^{7}+54720 \arctan \left (a x \right )^{2} x^{6} a^{6}+720 a^{6} x^{6}-12360 \arctan \left (a x \right ) x^{5} a^{5}+43200 \arctan \left (a x \right )^{2} x^{4} a^{4}+1608 a^{4} x^{4}-6150 \arctan \left (a x \right ) x^{3} a^{3}+2880 \arctan \left (a x \right )^{2} x^{2} a^{2}-94 a^{2} x^{2}+6345 \arctan \left (a x \right ) x a -5760 \arctan \left (a x \right )^{2}-6157\right )}{181440 a^{4}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{4032 a^{4} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}\,{d x} \]
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 \[ \int x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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 \[ \int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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