Integrand size = 14, antiderivative size = 91 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 5331
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=-2 a b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a^2 \log (c x)+i a b \left (\csc ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{24} i b^2 \left (\pi ^3-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-24 \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (127 ) = 254\).
Time = 1.09 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.71
method | result | size |
parts | \(a^{2} \ln \left (x \right )+b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 a b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(338\) |
derivativedivides | \(a^{2} \ln \left (c x \right )+b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 a b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(340\) |
default | \(a^{2} \ln \left (c x \right )+b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+2 a b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(340\) |
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x}\, dx \]
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\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x} \,d x \]
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