Optimal. Leaf size=123 \[ \frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}+\frac{\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac{\csc ^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.194174, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5259, 4427, 3785, 3919, 3831, 2660, 618, 204} \[ \frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}+\frac{\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac{\csc ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5259
Rule 4427
Rule 3785
Rule 3919
Rule 3831
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\csc ^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \operatorname{Subst}\left (\int \frac{x \cot (x) \csc (x)}{(-a+\csc (x))^3} \, dx,x,\csc ^{-1}(a+b x)\right )\right )\\ &=-\frac{\csc ^{-1}(a+b x)}{2 x^2}+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{1}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac{\csc ^{-1}(a+b x)}{2 x^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1-a^2-a \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac{\csc ^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1-2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{\csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac{\csc ^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1-2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac{\csc ^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1-2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x+x^2} \, dx,x,\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac{\csc ^{-1}(a+b x)}{2 x^2}+\frac{\left (2 \left (1-2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1-a^2\right )-x^2} \, dx,x,-2 a+2 \tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )}\\ &=-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac{\csc ^{-1}(a+b x)}{2 x^2}+\frac{\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac{a-\tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.741186, size = 199, normalized size = 1.62 \[ \frac{\frac{b x (a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}}{a \left (a^2-1\right )}+\frac{i \left (2 a^2-1\right ) b^2 x^2 \log \left (\frac{4 (a-1) a^2 (a+1) \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac{i \left (a^2+a b x-1\right )}{\sqrt{1-a^2}}\right )}{\left (2 a^2-1\right ) b^2 x}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{b^2 x^2 \sin ^{-1}\left (\frac{1}{a+b x}\right )}{a^2}-\csc ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.263, size = 453, normalized size = 3.7 \begin{align*} -{\frac{{\rm arccsc} \left (bx+a\right )}{2\,{x}^{2}}}+{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ) \left ({a}^{2}-1 \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{-1+ \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{{a}^{2}{b}^{2}}{bx+a}\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}-1}\sqrt{-1+ \left ( bx+a \right ) ^{2}}+a \left ( bx+a \right ) -1}{bx}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}} \left ({a}^{2}-1 \right ) ^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ){a}^{2} \left ({a}^{2}-1 \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{-1+ \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}+{\frac{b \left ( -1+ \left ( bx+a \right ) ^{2} \right ) }{ \left ( 2\,bx+2\,a \right ) a \left ({a}^{2}-1 \right ) x}{\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}+{\frac{3\,{b}^{2}}{2\,bx+2\,a}\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}-1}\sqrt{-1+ \left ( bx+a \right ) ^{2}}+a \left ( bx+a \right ) -1}{bx}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}} \left ({a}^{2}-1 \right ) ^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ){a}^{2}}\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}-1}\sqrt{-1+ \left ( bx+a \right ) ^{2}}+a \left ( bx+a \right ) -1}{bx}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}} \left ({a}^{2}-1 \right ) ^{-{\frac{5}{2}}}} \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{x^{2} \int \frac{{\left (b^{2} x + a b\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (b x + a - 1\right )\right )}}{b^{2} x^{4} + 2 \, a b x^{3} +{\left (b^{2} x^{4} + 2 \, a b x^{3} +{\left (a^{2} - 1\right )} x^{2}\right )}{\left (b x + a + 1\right )}{\left (b x + a - 1\right )} +{\left (a^{2} - 1\right )} x^{2}}\,{d x} + \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.46188, size = 998, normalized size = 8.11 \begin{align*} \left [\frac{{\left (2 \, a^{2} - 1\right )} \sqrt{a^{2} - 1} b^{2} x^{2} \log \left (\frac{a^{2} b x + a^{3} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{2} - \sqrt{a^{2} - 1} a - 1\right )} -{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1} - a}{x}\right ) - 2 \,{\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{3} - a\right )} b x -{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname{arccsc}\left (b x + a\right )}{2 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac{2 \,{\left (2 \, a^{2} - 1\right )} \sqrt{-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac{\sqrt{-a^{2} + 1} b x - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt{-a^{2} + 1}}{a^{2} - 1}\right ) - 2 \,{\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (a^{3} - a\right )} b x -{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname{arccsc}\left (b x + a\right )}{2 \,{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\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 \frac{\operatorname{acsc}{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.54176, size = 374, normalized size = 3.04 \begin{align*} b{\left (\frac{{\left (2 \, a^{2} b^{5} - b^{5}\right )} \arctan \left (-\frac{x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt{-a^{2} + 1}}\right )}{{\left (a^{4} b^{4} \mathrm{sgn}\left (b x + a\right ) - a^{2} b^{4} \mathrm{sgn}\left (b x + a\right )\right )} \sqrt{-a^{2} + 1}} - \frac{{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a b^{5} + a^{2} b^{4}{\left | b \right |} - b^{4}{\left | b \right |}}{{\left (a^{3} b^{4} \mathrm{sgn}\left (b x + a\right ) - a b^{4} \mathrm{sgn}\left (b x + a\right )\right )}{\left ({\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{2} - a^{2} + 1\right )}} - \frac{b \arctan \left (-\frac{{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} b + a{\left | b \right |}}{b}\right )}{a^{2} \mathrm{sgn}\left (b x + a\right )}\right )} - \frac{\arcsin \left (\frac{1}{b x + a}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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