Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}\right )}{b n}+\frac{\left (a+b x^n\right ) \text{csch}^{-1}\left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.0702649, antiderivative size = 46, 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, 6314, 372, 266, 63, 207} \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}\right )}{b n}+\frac{\left (a+b x^n\right ) \text{csch}^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 6314
Rule 372
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int x^{-1+n} \text{csch}^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right ) \text{csch}^{-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 ) \text{csch}^{-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 ) \text{csch}^{-1}\left (a+b x^n\right )}{b n}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{\left (a+b x^n\right )^2}\right )}{2 b n}\\ &=\frac{\left (a+b x^n\right ) \text{csch}^{-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 ) \text{csch}^{-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 [A] time = 0.173554, size = 74, normalized size = 1.61 \[ \frac{\frac{\sqrt{\left (a+b x^n\right )^2+1} \sinh ^{-1}\left (a+b x^n\right )}{\sqrt{\frac{1}{\left (a+b x^n\right )^2}+1}}+\left (a+b x^n\right )^2 \text{csch}^{-1}\left (a+b x^n\right )}{b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}{\rm arccsch} \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 = 1.03147, size = 81, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )} \operatorname{arcsch}\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 = 3.01241, size = 969, normalized size = 21.07 \begin{align*} \frac{a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1\right ) - a \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} - 1\right ) +{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac{\sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}} + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a}\right ) - \log \left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right ) - a + \sqrt{\frac{2 \, a b +{\left (a^{2} + b^{2} + 1\right )} \cosh \left (n \log \left (x\right )\right ) -{\left (a^{2} - b^{2} + 1\right )} \sinh \left (n \log \left (x\right )\right )}{\cosh \left (n \log \left (x\right )\right ) - \sinh \left (n \log \left (x\right )\right )}}\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{arcsch}\left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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