Optimal. Leaf size=464 \[ \frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.40, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5259, 4427, 4190, 4183, 2531, 2282, 6589, 4184, 3717, 2190, 2279, 2391, 4186, 3770} \[ -\frac {6 i a^2 \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3717
Rule 3770
Rule 4183
Rule 4184
Rule 4186
Rule 4190
Rule 4427
Rule 5259
Rule 6589
Rubi steps
\begin {align*} \int x^2 \csc ^{-1}(a+b x)^3 \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \cot (x) \csc (x) (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\operatorname {Subst}\left (\int x^2 (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\operatorname {Subst}\left (\int \left (-a^3 x^2+3 a^2 x^2 \csc (x)-3 a x^2 \csc ^2(x)+x^2 \csc ^3(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\operatorname {Subst}\left (\int x^2 \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(3 a) \operatorname {Subst}\left (\int x^2 \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^3}-\frac {\operatorname {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(6 a) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\operatorname {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {(12 i a) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 i a^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (6 i a^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {(6 a) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(3 i a) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 8.89, size = 656, normalized size = 1.41 \[ -\frac {24 i \left (6 a^2+1\right ) \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )-24 i \left (6 a^2+1\right ) \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )-144 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )+144 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )-12 a^2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-12 a^2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+72 i a \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-24 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )+24 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )+72 i a \csc ^{-1}(a+b x)^2-\frac {\csc ^{-1}(a+b x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+6 a \csc ^{-1}(a+b x)^3 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-3 \csc ^{-1}(a+b x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+36 a \csc ^{-1}(a+b x)^2 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 \csc ^{-1}(a+b x) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+12 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-12 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-144 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-36 a \csc ^{-1}(a+b x)^2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 \csc ^{-1}(a+b x) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+24 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )}{24 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arccsc}\left (b x + a\right )^{3}, x\right ) \]
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 \[ \int x^{2} \operatorname {arccsc}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.72, size = 809, normalized size = 1.74 \[ -\frac {6 i a \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {i \mathrm {arccsc}\left (b x +a \right ) \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {2 \mathrm {arccsc}\left (b x +a \right )^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, x a}{b^{2}}+\frac {i \mathrm {arccsc}\left (b x +a \right ) \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {6 i a \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {\mathrm {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2 b^{3}}+\frac {\mathrm {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2 b^{3}}+\frac {6 a^{2} \polylog \left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {6 a^{2} \polylog \left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {\mathrm {arccsc}\left (b x +a \right ) x}{b^{2}}+\frac {\mathrm {arccsc}\left (b x +a \right ) a}{b^{3}}-\frac {5 \mathrm {arccsc}\left (b x +a \right )^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, a^{2}}{2 b^{3}}+\frac {\mathrm {arccsc}\left (b x +a \right )^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, x^{2}}{2 b}+\frac {a^{3} \mathrm {arccsc}\left (b x +a \right )^{3}}{3 b^{3}}-\frac {3 i a \mathrm {arccsc}\left (b x +a \right )^{2}}{b^{3}}+\frac {6 a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {6 a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {3 a^{2} \mathrm {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {3 a^{2} \mathrm {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {6 i a^{2} \mathrm {arccsc}\left (b x +a \right ) \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {6 i a^{2} \mathrm {arccsc}\left (b x +a \right ) \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {2 \arctanh \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {\polylog \left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {\polylog \left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {x^{3} \mathrm {arccsc}\left (b x +a \right )^{3}}{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 \[ \frac {1}{3} \, x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{3} - \frac {1}{4} \, x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} - \int \frac {12 \, {\left (b^{3} x^{5} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + 3 \, a b^{2} x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + {\left (3 \, a^{2} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )\right )} b x^{3} + {\left (a^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - a \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )\right )} x^{2}\right )} \log \left (b x + a\right )^{2} - {\left (4 \, b x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{2} - b x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 4 \, {\left (b^{3} x^{5} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + 2 \, a b^{2} x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + {\left (a^{2} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )\right )} b x^{3} + 3 \, {\left (b^{3} x^{5} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + 3 \, a b^{2} x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + {\left (3 \, a^{2} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )\right )} b x^{3} + {\left (a^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - a \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )\right )} x^{2}\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{4 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{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 \[ \int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \]
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 \[ \int x^{2} \operatorname {acsc}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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