Optimal. Leaf size=48 \[ \frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
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Rubi [A] time = 0.0697476, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6715, 5251, 372, 266, 63, 206} \[ \frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 5251
Rule 372
Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \csc ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}} \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{1}{x^2}} x} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{\left (a+b x^n\right )^2}\right )}{2 b n}\\ &=\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{\left (a+b x^n\right )^2}}\right )}{b n}\\ \end{align*}
Mathematica [B] time = 0.30678, size = 130, normalized size = 2.71 \[ \frac{\sqrt{\left (a+b x^n\right )^2-1} \left (\log \left (\frac{a+b x^n}{\sqrt{\left (a+b x^n\right )^2-1}}+1\right )-\log \left (1-\frac{a+b x^n}{\sqrt{\left (a+b x^n\right )^2-1}}\right )\right )}{2 b n \left (a+b x^n\right ) \sqrt{1-\frac{1}{\left (a+b x^n\right )^2}}}+\frac{\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arccsc} \left (a+b{x}^{n}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98776, size = 89, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )} \operatorname{arccsc}\left (b x^{n} + a\right ) + \log \left (\sqrt{-\frac{1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt{-\frac{1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.27864, size = 216, normalized size = 4.5 \begin{align*} \frac{b x^{n} \operatorname{arccsc}\left (b x^{n} + a\right ) - 2 \, a \arctan \left (-b x^{n} - a + \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right ) - \log \left (-b x^{n} - a + \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{n - 1} \operatorname{arccsc}\left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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