Optimal. Leaf size=531 \[ \frac {c x \sqrt {a^2 c x^2+c}}{36 a^2}-\frac {19 c x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac {c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{15} a c x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {1}{60} c x^3 \sqrt {a^2 c x^2+c}+\frac {7}{24} c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {41 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a^3}-\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {31 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3} \]
[Out]
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Rubi [A] time = 3.19, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 92, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ -\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {41 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a^3}+\frac {1}{60} c x^3 \sqrt {a^2 c x^2+c}+\frac {c x \sqrt {a^2 c x^2+c}}{36 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{15} a c x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {7}{24} c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {19 c x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac {c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac {31 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 321
Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4930
Rule 4950
Rule 4952
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}+2 \left (\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\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)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} \left (5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {1}{\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)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}\right )+\frac {1}{15} \left (4 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^2\right ) \int \frac {x^4}{\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)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a^2}-\frac {\left (8 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \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}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}+\frac {\left (8 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{12 a^2}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{3 a^2}-\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{4 a^2}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+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 ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}+\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{72 a^2}+\frac {\left (8 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{18 a^2}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{8 a^2}+\frac {\left (i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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 (i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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 ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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 (3 i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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 (5 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \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 (5 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \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 (c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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 (5 i c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \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}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \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 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\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)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {13 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\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)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\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 [A] time = 3.67, size = 527, normalized size = 0.99 \[ \frac {c \sqrt {a^2 c x^2+c} \left (960 \left (-2 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-3 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+3 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+3 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-3 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+3 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+32 \left (19 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+45 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-45 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )-45 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+45 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )-45 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+\left (a^2 x^2+1\right )^3 \left (-\frac {56 a x}{\sqrt {a^2 x^2+1}}+15 \tan ^{-1}(a x)^2 \left (\frac {78 a x}{\sqrt {a^2 x^2+1}}-47 \sin \left (3 \tan ^{-1}(a x)\right )+3 \sin \left (5 \tan ^{-1}(a x)\right )\right )+\tan ^{-1}(a x) \left (\frac {12}{\sqrt {a^2 x^2+1}}+110 \cos \left (3 \tan ^{-1}(a x)\right )-90 \cos \left (5 \tan ^{-1}(a x)\right )\right )-108 \sin \left (3 \tan ^{-1}(a x)\right )-52 \sin \left (5 \tan ^{-1}(a x)\right )\right )+120 \left (a^2 x^2+1\right )^{3/2} \left (-3 \tan ^{-1}(a x)^2 \left (\sqrt {a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )-7 a x\right )+2 \left (\sqrt {a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )+a x\right )+\tan ^{-1}(a x) \left (6 \sqrt {a^2 x^2+1} \cos \left (3 \tan ^{-1}(a x)\right )+2\right )\right )\right )}{11520 a^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.53, size = 338, normalized size = 0.64 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 \arctan \left (a x \right )^{2} x^{5} a^{5}-48 \arctan \left (a x \right ) x^{4} a^{4}+210 \arctan \left (a x \right )^{2} x^{3} a^{3}+12 a^{3} x^{3}-76 \arctan \left (a x \right ) a^{2} x^{2}+45 \arctan \left (a x \right )^{2} x a +20 a x +62 \arctan \left (a x \right )\right )}{720 a^{3}}+\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (45 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+164 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a^{3} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{2}\,{d x} \]
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 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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