Optimal. Leaf size=919 \[ -\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {a^2 c x^2+c}}+a^2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac {c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}+\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 c x^2+c}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {3 a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \tanh ^{-1}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.02, antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4950, 4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181} \[ -\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {a^2 c x^2+c}}+a^2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac {c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}+\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 c x^2+c}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {3 a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \tanh ^{-1}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4181
Rule 4183
Rule 4888
Rule 4890
Rule 4930
Rule 4944
Rule 4950
Rule 4954
Rule 4956
Rule 4958
Rule 4962
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x^3} \, dx &=c \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)^3}{x^3 \sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx-\left (3 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\left (3 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+2 \frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}\right )-\frac {\left (3 a^2 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 a^2 c^2 \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 )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c^2 \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 )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 a^2 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^2 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 )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^2 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^2 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 )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}\right )-\frac {\left (3 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^2 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^2 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 )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a^2 c^2 \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 )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3 a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 x^2}+\frac {a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 9.52, size = 691, normalized size = 0.75 \[ \frac {a^2 c \sqrt {a^2 c x^2+c} \tan \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \left (72 i \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-96 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+96 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+144 \tan ^{-1}(a x) \text {Li}_3\left (e^{-i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-144 \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+24 i \left (3 \tan ^{-1}(a x)^2+2\right ) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-48 i \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+96 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-96 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-144 i \text {Li}_4\left (e^{-i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-144 i \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x)^2-12 \tan ^{-1}(a x)^2 \cot ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+6 i \tan ^{-1}(a x)^4 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-3 i \pi ^4 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+8 a x \tan ^{-1}(a x)^3 \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-24 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-48 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+48 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+48 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-48 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-2 \tan ^{-1}(a x)^3 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+2 \tan ^{-1}(a x)^3 \csc \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )}{16 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}}, x\right ) \]
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 = 1.70, size = 592, normalized size = 0.64 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (2 \arctan \left (a x \right ) a^{2} x^{2}-3 a x -\arctan \left (a x \right )\right )}{2 x^{2}}+\frac {3 a^{2} c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right )^{2} \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 a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \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 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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