Optimal. Leaf size=553 \[ \frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {202 a x}{27 c^2 \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {a^2 c x^2+c}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {2 a x}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.92, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {4966, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4898, 191, 4900, 192} \[ \frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {202 a x}{27 c^2 \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {a^2 c x^2+c}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {2 a x}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 2282
Rule 2531
Rule 4183
Rule 4898
Rule 4900
Rule 4930
Rule 4956
Rule 4958
Rule 4966
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-a \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx}{c^2}-\frac {a^2 \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}+\frac {1}{9} (2 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx-\frac {(2 a) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {(3 a) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}+\frac {(4 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}+\frac {(4 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac {(6 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 347, normalized size = 0.63 \[ \frac {\left (a^2 x^2+1\right )^{3/2} \left (\frac {1620 a x}{\sqrt {a^2 x^2+1}}+\frac {270 \tan ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}-\frac {810 a x \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}-\frac {1620 \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+648 i \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \tan ^{-1}(a x)}\right )+648 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )+1296 \tan ^{-1}(a x) \text {Li}_3\left (e^{-i \tan ^{-1}(a x)}\right )-1296 \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )-1296 i \text {Li}_4\left (e^{-i \tan ^{-1}(a x)}\right )-1296 i \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )+54 i \tan ^{-1}(a x)^4+216 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )-216 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-18 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \sin \left (3 \tan ^{-1}(a x)\right )+18 \tan ^{-1}(a x)^3 \cos \left (3 \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-27 i \pi ^4\right )}{216 c \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}, x\right ) \]
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 = 0.88, size = 560, normalized size = 1.01 \[ -\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3}}+\frac {5 \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3}}+\frac {i \left (i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}} \]
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 \[ \int \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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