Integrand size = 12, antiderivative size = 464 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5367, 4512, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438, 4271, 3855} \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3 \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3798
Rule 3855
Rule 4268
Rule 4269
Rule 4271
Rule 4275
Rule 4512
Rule 5367
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {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 {\text {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 {\text {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 {\text {Subst}\left (\int x^2 \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(3 a) \text {Subst}\left (\int x^2 \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \text {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 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^3}-\frac {\text {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(6 a) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 a^2\right ) \text {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 ) \text {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 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\text {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) \text {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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (6 i a^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,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 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {(6 a) \text {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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(3 i a) \text {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 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\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) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \\ \end{align*}
Time = 7.18 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.41 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=-\frac {72 i a \csc ^{-1}(a+b x)^2-12 \csc ^{-1}(a+b x) \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 )-2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \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 )-3 \csc ^{-1}(a+b x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {\csc ^{-1}(a+b x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+12 \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 \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 )-144 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )+24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+72 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-144 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+24 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+144 a^2 \operatorname {PolyLog}\left (3,e^{i \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 )+6 a \csc ^{-1}(a+b x)^3 \sec ^2\left (\frac {1}{2} \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 )-12 \csc ^{-1}(a+b x) \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 )-2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \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 )}{24 b^3} \]
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Time = 1.49 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.61
method | result | size |
derivativedivides | \(\frac {-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\frac {\operatorname {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}-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-\operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {\operatorname {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}-i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+6 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i a \operatorname {arccsc}\left (b x +a \right )^{2}+6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-3 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (6 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}-18 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i a \,\operatorname {arccsc}\left (b x +a \right )+6 b x +6 a \right )}{6}+3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}-6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}}{b^{3}}\) | \(749\) |
default | \(\frac {-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\frac {\operatorname {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}-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-\operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {\operatorname {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}-i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+6 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i a \operatorname {arccsc}\left (b x +a \right )^{2}+6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-3 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (6 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}-18 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i a \,\operatorname {arccsc}\left (b x +a \right )+6 b x +6 a \right )}{6}+3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}-6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}}{b^{3}}\) | \(749\) |
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\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
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\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int x^{2} \operatorname {acsc}^{3}{\left (a + b x \right )}\, dx \]
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\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
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\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \]
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