Optimal. Leaf size=272 \[ -\frac{2 i a^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac{2 i a^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac{i \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac{i \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac{a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac{4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac{2 a \log (a+b x)}{b^3}-\frac{2 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac{2 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac{1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac{x}{3 b^2} \]
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Rubi [A] time = 0.230817, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, 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 5259
Rule 4427
Rule 4190
Rule 4183
Rule 2279
Rule 2391
Rule 4184
Rule 3475
Rule 4185
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*}
Mathematica [A] time = 4.66081, size = 314, normalized size = 1.15 \[ -\frac{8 \left (6 a^2+1\right ) \left (i \left (\text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )+\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )\right )-2 \left (\left (6 a^2+1\right ) \csc ^{-1}(a+b x)^2-12 a \csc ^{-1}(a+b x)+2\right ) \cot \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-2 \left (\left (6 a^2+1\right ) \csc ^{-1}(a+b x)^2+12 a \csc ^{-1}(a+b x)+2\right ) \tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-48 a \log \left (\frac{1}{a+b x}\right )-\frac{\csc ^{-1}(a+b x)^2 \csc ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+2 \csc ^{-1}(a+b x) \left (3 a \csc ^{-1}(a+b x)-1\right ) \csc ^2\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^2 \sin ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )+2 \csc ^{-1}(a+b x) \left (3 a \csc ^{-1}(a+b x)+1\right ) \sec ^2\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.588, size = 545, normalized size = 2. \begin{align*}{\frac{a}{3\,{b}^{3}}}+{\frac{x}{3\,{b}^{2}}}+{\frac{{\frac{i}{3}}}{{b}^{3}}{\it polylog} \left ( 2,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-{\frac{{\frac{i}{3}}}{{b}^{3}}{\it polylog} \left ( 2,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+2\,{\frac{a}{{b}^{3}}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-4\,{\frac{a}{{b}^{3}}\ln \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{{x}^{2}{\rm arccsc} \left (bx+a\right )}{3\,b}\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{x}^{3} \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{3}}-{\frac{5\,{\rm arccsc} \left (bx+a\right ){a}^{2}}{3\,{b}^{3}}\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}+2\,{\frac{a}{{b}^{3}}\ln \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}}-1 \right ) }-{\frac{{\rm arccsc} \left (bx+a\right )}{3\,{b}^{3}}\ln \left ( 1-{\frac{i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{{\rm arccsc} \left (bx+a\right )}{3\,{b}^{3}}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{{a}^{3} \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{3\,{b}^{3}}}-{\frac{4\,x{\rm arccsc} \left (bx+a\right )a}{3\,{b}^{2}}\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{2\,ia{\rm arccsc} \left (bx+a\right )}{{b}^{3}}}+{\frac{2\,i{a}^{2}}{{b}^{3}}{\it polylog} \left ( 2,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-{\frac{2\,i{a}^{2}}{{b}^{3}}{\it polylog} \left ( 2,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-2\,{\frac{{\rm arccsc} \left (bx+a\right ){a}^{2}}{{b}^{3}}\ln \left ( 1-{\frac{i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+2\,{\frac{{\rm arccsc} \left (bx+a\right ){a}^{2}}{{b}^{3}}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{arccsc}\left (b x + a\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acsc}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arccsc}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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