Optimal. Leaf size=747 \[ \frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2}+\frac {1}{4} x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{4 a^3}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt {a^2 c x^2+c}} \]
[Out]
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Rubi [A] time = 1.85, antiderivative size = 747, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261} \[ -\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{4 a^3}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {1}{4} x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt {a^2 c x^2+c}}+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2282
Rule 2531
Rule 4181
Rule 4886
Rule 4888
Rule 4890
Rule 4930
Rule 4950
Rule 4952
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx &=c \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{4} (3 c) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {(3 c) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}-\frac {1}{4} (3 a c) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{2} c \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {(3 c) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {(3 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {(9 c) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {\left (c \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 {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {(9 c) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \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}}+\frac {\left (3 c \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 {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 \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 \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 \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 \sqrt {1+a^2 x^2}\right ) \operatorname {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 (3 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c \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 \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 \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 {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 {i c \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 \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 \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 {i c \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 {i c \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 (3 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 {i c \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 \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 {3 i c \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 {i c \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 {i c \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 {3 c \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 \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 (9 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 {i c \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 \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 {3 i c \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 {i c \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 {i c \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 {3 c \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 {3 c \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 (3 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 {i c \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 \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 {3 i c \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 {i c \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 {i c \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 {3 c \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 {3 c \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 {3 i c \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 \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 {\left (9 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c \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 {i c \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 \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 {3 i c \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 {i c \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 {i c \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 {3 c \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 {3 c \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 {3 i c \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 {3 i c \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}}\\ \end {align*}
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Mathematica [B] time = 12.13, size = 1844, normalized size = 2.47 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.80, size = 460, normalized size = 0.62 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{3} x^{3}-2 \arctan \left (a x \right )^{2} x^{2} a^{2}+\arctan \left (a x \right )^{3} x a +2 \arctan \left (a x \right ) x a +\arctan \left (a x \right )^{2}-2\right )}{8 a^{3}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a^{3} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3}\,{d x} \]
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 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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 \[ \int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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