Optimal. Leaf size=476 \[ \frac {c x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac {1}{7} a^2 c x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{21} a c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {8}{35} c x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {23 c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac {17 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{140 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{105 a^4 c}-\frac {17 \left (a^2 c x^2+c\right )^{3/2}}{1260 a^4}-\frac {17 c \sqrt {a^2 c x^2+c}}{280 a^4}-\frac {2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4}+\frac {3 c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3} \]
[Out]
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Rubi [A] time = 4.07, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 75, 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} \[ \frac {17 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {17 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{140 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{105 a^4 c}-\frac {17 \left (a^2 c x^2+c\right )^{3/2}}{1260 a^4}-\frac {17 c \sqrt {a^2 c x^2+c}}{280 a^4}+\frac {1}{7} a^2 c x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{21} a c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {8}{35} c x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {23 c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac {c x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac {3 c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}-\frac {2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4} \]
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 )^{3/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (2 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+2 \left (\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{5} \left (4 c^2\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^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{7} \left (6 a^2 c^2\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^2\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{35} \left (24 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {\left (4 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c^2 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+2 \left (-\frac {c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{10} c^2 \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (8 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {\left (8 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}\right )+\frac {1}{21} \left (5 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (12 a c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{21} \left (a^2 c^2\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c \sqrt {c+a^2 c x^2}}{3 a^4}-\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{84} \left (5 c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (3 c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{20} c^2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {\left (4 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (16 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {\left (3 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {\left (4 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}\right )-\frac {\left (16 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{28 a}-\frac {\left (9 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {\left (16 c^2\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^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (c^2 \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^2 \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 \sqrt {c+a^2 c x^2}}{3 a^4}-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {10 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 )}{3 a^4 \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 )}{3 a^4 \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 )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {1}{168} \left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{70} \left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^3}+\frac {\left (9 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^3}+\frac {\left (8 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (32 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^3}+\frac {\left (5 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{56 a^2}+\frac {\left (9 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{70 a^2}+\frac {\left (8 c^2\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}+\frac {1}{42} \left (a^2 c^2\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 )+2 \left (-\frac {5 c \sqrt {c+a^2 c x^2}}{12 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{20} c^2 \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 (3 c^2 \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^2 \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^2 \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 {139 c \sqrt {c+a^2 c x^2}}{168 a^4}-\frac {2 \left (c+a^2 c x^2\right )^{3/2}}{63 a^4}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^4 c}-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {10 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 )}{3 a^4 \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 )}{3 a^4 \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 )}{3 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {31 c \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {89 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 )}{30 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}\right )-\frac {1}{168} \left (5 c^2\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^2\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^2 \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^2 \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^2 \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^2 \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 {817 c \sqrt {c+a^2 c x^2}}{840 a^4}-\frac {101 \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 c}-\frac {131 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{168 a^3}+\frac {61 c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{420 a}-\frac {1}{21} a c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {118 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^4}+\frac {59 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{105 a^2}-\frac {6}{35} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {2543 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 )}{420 a^4 \sqrt {c+a^2 c x^2}}+\frac {2543 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}-\frac {2543 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 )}{840 a^4 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {31 c \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4}+\frac {5 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10 a}+\frac {8 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^4}-\frac {4 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {89 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 )}{30 a^4 \sqrt {c+a^2 c x^2}}-\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}+\frac {89 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 )}{60 a^4 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 4.81, size = 797, normalized size = 1.67 \[ \frac {c \left (a^2 x^2+1\right )^2 \sqrt {a^2 c x^2+c} \left (\left (a^2 x^2+1\right ) \left (-5376 \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2+6720 \cos \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2+10944 \tan ^{-1}(a x)^2-6489 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-2163 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-309 \cos \left (7 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac {10815 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+6489 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+2163 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+309 \cos \left (7 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {10815 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-1266 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+360 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-618 \sin \left (6 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+6262 \cos \left (2 \tan ^{-1}(a x)\right )+2764 \cos \left (4 \tan ^{-1}(a x)\right )+618 \cos \left (6 \tan ^{-1}(a x)\right )-\frac {19776 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{7/2}}+\frac {19776 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{7/2}}+4116\right )-168 \left (160 \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac {110 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {110 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+4 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-22 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+72 \cos \left (2 \tan ^{-1}(a x)\right )+22 \cos \left (4 \tan ^{-1}(a x)\right )-\frac {176 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+\frac {176 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+50\right )\right )}{161280 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{5} + c x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 2.28, size = 271, normalized size = 0.57 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (360 \arctan \left (a x \right )^{2} x^{6} a^{6}-120 \arctan \left (a x \right ) x^{5} a^{5}+576 \arctan \left (a x \right )^{2} x^{4} a^{4}+24 a^{4} x^{4}-138 \arctan \left (a x \right ) x^{3} a^{3}+72 \arctan \left (a x \right )^{2} x^{2} a^{2}+14 a^{2} x^{2}+135 \arctan \left (a x \right ) x a -144 \arctan \left (a x \right )^{2}-163\right )}{2520 a^{4}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{280 a^{4} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3} \arctan \left (a x\right )^{2}\,{d x} \]
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 \[ \int x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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