Optimal. Leaf size=84 \[ x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}-\frac {2 i b^2 \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c} \]
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Rubi [A] time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5217, 4410, 4183, 2279, 2391} \[ -\frac {2 i b^2 \text {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \text {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4183
Rule 4410
Rule 5217
Rubi steps
\begin {align*} \int \left (a+b \csc ^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^2-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \csc ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 147, normalized size = 1.75 \[ \frac {a^2 c x+2 a b c x \csc ^{-1}(c x)+2 a b \log \left (\cos \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-2 a b \log \left (\sin \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-2 i b^2 \text {Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )+2 i b^2 \text {Li}_2\left (e^{i \csc ^{-1}(c x)}\right )+b^2 c x \csc ^{-1}(c x)^2-2 b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )+2 b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b \operatorname {arccsc}\left (c x\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 196, normalized size = 2.33 \[ x \,b^{2} \mathrm {arccsc}\left (c x \right )^{2}+2 x a b \,\mathrm {arccsc}\left (c x \right )-\frac {2 b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}+\frac {2 b^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}-\frac {2 i \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) b^{2}}{c}+\frac {2 i \dilog \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) b^{2}}{c}+a^{2} x +\frac {2 \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) a b}{c} \]
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 \[ -\frac {1}{4} \, {\left (2 \, c^{2} {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )} \log \relax (c)^{2} - 4 \, c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2}\right )}{c^{2} x^{2} - 1}\,{d x} \log \relax (c) + 8 \, c^{2} \int \frac {x^{2} \log \relax (x)}{c^{2} x^{2} - 1}\,{d x} \log \relax (c) - 4 \, x \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - 4 \, c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2}\right ) \log \relax (x)}{c^{2} x^{2} - 1}\,{d x} + 4 \, c^{2} \int \frac {x^{2} \log \relax (x)^{2}}{c^{2} x^{2} - 1}\,{d x} - 4 \, c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2}\right )}{c^{2} x^{2} - 1}\,{d x} + x \log \left (c^{2} x^{2}\right )^{2} + 2 \, {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \log \relax (c)^{2} + 4 \, \int \frac {\log \left (c^{2} x^{2}\right )}{c^{2} x^{2} - 1}\,{d x} \log \relax (c) - 8 \, \int \frac {\log \relax (x)}{c^{2} x^{2} - 1}\,{d x} \log \relax (c) - 8 \, \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} \arctan \left (\frac {1}{\sqrt {c x + 1} \sqrt {c x - 1}}\right )}{c^{2} x^{2} - 1}\,{d x} + 4 \, \int \frac {\log \left (c^{2} x^{2}\right ) \log \relax (x)}{c^{2} x^{2} - 1}\,{d x} - 4 \, \int \frac {\log \relax (x)^{2}}{c^{2} x^{2} - 1}\,{d x} + 4 \, \int \frac {\log \left (c^{2} x^{2}\right )}{c^{2} x^{2} - 1}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} a b}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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