Optimal. Leaf size=36 \[ \frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b} \]
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Rubi [A] time = 0.023543, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5251, 372, 266, 63, 206} \[ \frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5251
Rule 372
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \csc ^{-1}(a+b x) \, dx &=\frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\int \frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}} \, dx\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{1}{x^2}} x} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{(a+b x)^2}\right )}{2 b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.112826, size = 114, normalized size = 3.17 \[ \frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (\tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2-1}}\right )-a \tan ^{-1}\left (\sqrt{(a+b x)^2-1}\right )\right )}{b \sqrt{a^2+2 a b x+b^2 x^2-1}}+x \csc ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.236, size = 50, normalized size = 1.4 \begin{align*} x{\rm arccsc} \left (bx+a\right )+{\frac{{\rm arccsc} \left (bx+a\right )a}{b}}+{\frac{1}{b}\ln \left ( bx+a+ \left ( bx+a \right ) \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 [A] time = 0.996672, size = 74, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (b x + a\right )} \operatorname{arccsc}\left (b x + a\right ) + \log \left (\sqrt{-\frac{1}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt{-\frac{1}{{\left (b x + a\right )}^{2}} + 1} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.06341, size = 184, normalized size = 5.11 \begin{align*} \frac{b x \operatorname{arccsc}\left (b x + a\right ) - 2 \, a \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{b} \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 \operatorname{acsc}{\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 \operatorname{arccsc}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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