Integrand size = 12, antiderivative size = 324 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5367, 4648, 4625, 3798, 2221, 2611, 2320, 6724, 4615} \[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4615
Rule 4625
Rule 4648
Rule 5367
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \cot (x) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\text {Subst}\left (\int \frac {x^2 \cot (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\left (a \text {Subst}\left (\int \frac {x^2 \cos (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\right )-\text {Subst}\left (\int x^2 \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = 2 i \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^{i x} x^2}{1-\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^{i x} x^2}{1+\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 \text {Subst}\left (\int x \log \left (1+\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-2 \text {Subst}\left (\int x \log \left (1+\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 \text {Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )+2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{-1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )+2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {i a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right ) \\ & = \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.26 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\frac {i \pi ^3}{6}-\frac {1}{3} i \csc ^{-1}(a+b x)^3-\csc ^{-1}(a+b x)^2 \log \left (1-e^{-i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x)^2 \log \left (1-\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{-i \csc ^{-1}(a+b x)}\right )+2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-2 \operatorname {PolyLog}\left (3,e^{-i \csc ^{-1}(a+b x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right ) \]
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\[\int \frac {\operatorname {arccsc}\left (b x +a \right )^{2}}{x}d x\]
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\[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\int \frac {\operatorname {acsc}^{2}{\left (a + b x \right )}}{x}\, dx \]
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\[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\csc ^{-1}(a+b x)^2}{x} \, dx=\int \frac {{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2}{x} \,d x \]
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