Optimal. Leaf size=272 \[ \frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]
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Rubi [A] time = 0.23, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5259, 4427, 4190, 4183, 2279, 2391, 4184, 3475, 4185} \[ -\frac {2 i a^2 \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4183
Rule 4184
Rule 4185
Rule 4190
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int x^2 \csc ^{-1}(a+b x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \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)^2-\frac {2 \operatorname {Subst}\left (\int x (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int \left (-a^3 x+3 a^2 x \csc (x)-3 a x \csc ^2(x)+x \csc ^3(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \operatorname {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}+\frac {\operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\left (2 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}\\ &=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 4.74, size = 314, normalized size = 1.15 \[ -\frac {8 \left (6 a^2+1\right ) \left (i \left (\text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )-\text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )\right )+\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )\right )-2 \left (\left (6 a^2+1\right ) \csc ^{-1}(a+b x)^2-12 a \csc ^{-1}(a+b x)+2\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \left (\left (6 a^2+1\right ) \csc ^{-1}(a+b x)^2+12 a \csc ^{-1}(a+b x)+2\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-48 a \log \left (\frac {1}{a+b x}\right )-\frac {\csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+2 \csc ^{-1}(a+b x) \left (3 a \csc ^{-1}(a+b x)-1\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^2 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \csc ^{-1}(a+b x) \left (3 a \csc ^{-1}(a+b x)+1\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arccsc}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arccsc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.76, size = 545, normalized size = 2.00 \[ -\frac {4 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x a}{3 b^{2}}+\frac {i \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{3}}-\frac {2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {2 a^{2} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {x^{3} \mathrm {arccsc}\left (b x +a \right )^{2}}{3}-\frac {5 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) a^{2}}{3 b^{3}}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x^{2}}{3 b}+\frac {2 i a \,\mathrm {arccsc}\left (b x +a \right )}{b^{3}}+\frac {2 i a^{2} \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}+\frac {2 a \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}}-\frac {4 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a}{b^{3}}+\frac {2 a \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )}{b^{3}}+\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{3}}-\frac {\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{3}}+\frac {a^{3} \mathrm {arccsc}\left (b x +a \right )^{2}}{3 b^{3}}+\frac {a}{3 b^{3}}+\frac {x}{3 b^{2}}-\frac {i \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{3}}-\frac {2 i a^{2} \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{3}} \]
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 \[ \frac {1}{3} \, x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{2} - \frac {1}{12} \, x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \int \frac {2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} b x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - 3 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} - 1\right )} b x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{5} + 2 \, a b^{2} x^{4} + {\left (a^{2} - 1\right )} b x^{3} + 3 \, {\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} + {\left (3 \, a^{2} - 1\right )} b x^{3} + {\left (a^{3} - a\right )} x^{2}\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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