Optimal. Leaf size=91 \[ i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{2} b^2 \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5223, 3717, 2190, 2531, 2282, 6589} \[ i b \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{2} b^2 \text {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 5223
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx &=-\operatorname {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 \operatorname {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) \operatorname {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 ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\left (i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (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 ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_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 ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 137, normalized size = 1.51 \[ a^2 \log (c x)+i a b \left (\csc ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\right )-2 a b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{24} i b^2 \left (-24 \csc ^{-1}(c x) \text {Li}_2\left (e^{-2 i \csc ^{-1}(c x)}\right )+12 i \text {Li}_3\left (e^{-2 i \csc ^{-1}(c x)}\right )-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )+\pi ^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b \operatorname {arccsc}\left (c x\right ) + a^{2}}{x}, x\right ) \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 361, normalized size = 3.97 \[ a^{2} \ln \left (c x \right )+\frac {i b^{2} \mathrm {arccsc}\left (c x \right )^{3}}{3}-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i a b \mathrm {arccsc}\left (c x \right )^{2}-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \]
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 \[ -\frac {1}{2} \, b^{2} c^{2} {\left (\frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \log \relax (c)^{2} + b^{2} c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2}\right )}{c^{2} x^{3} - x}\,{d x} \log \relax (c) - 2 \, b^{2} c^{2} \int \frac {x^{2} \log \relax (x)}{c^{2} x^{3} - x}\,{d x} \log \relax (c) + 2 \, b^{2} c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2}\right ) \log \relax (x)}{c^{2} x^{3} - x}\,{d x} - b^{2} c^{2} \int \frac {x^{2} \log \relax (x)^{2}}{c^{2} x^{3} - x}\,{d x} + 2 \, a b c^{2} \int \frac {x^{2} \arctan \left (\frac {1}{\sqrt {c x + 1} \sqrt {c x - 1}}\right )}{c^{2} x^{3} - x}\,{d x} + \frac {1}{2} \, b^{2} {\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \relax (x)\right )} \log \relax (c)^{2} + b^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} \log \relax (x) - \frac {1}{4} \, b^{2} \log \left (c^{2} x^{2}\right )^{2} \log \relax (x) - b^{2} \int \frac {\log \left (c^{2} x^{2}\right )}{c^{2} x^{3} - x}\,{d x} \log \relax (c) + 2 \, b^{2} \int \frac {\log \relax (x)}{c^{2} x^{3} - x}\,{d x} \log \relax (c) + 2 \, b^{2} \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} \arctan \left (\frac {1}{\sqrt {c x + 1} \sqrt {c x - 1}}\right ) \log \relax (x)}{c^{2} x^{3} - x}\,{d x} - 2 \, b^{2} \int \frac {\log \left (c^{2} x^{2}\right ) \log \relax (x)}{c^{2} x^{3} - x}\,{d x} + b^{2} \int \frac {\log \relax (x)^{2}}{c^{2} x^{3} - x}\,{d x} - 2 \, a b \int \frac {\arctan \left (\frac {1}{\sqrt {c x + 1} \sqrt {c x - 1}}\right )}{c^{2} x^{3} - x}\,{d x} + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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