Optimal. Leaf size=47 \[ \frac{2 \sqrt{a+b \sinh ^n(x)}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{n} \]
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Rubi [A] time = 0.0919908, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3230, 266, 50, 63, 208} \[ \frac{2 \sqrt{a+b \sinh ^n(x)}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 3230
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \coth (x) \sqrt{a+b \sinh ^n(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^n}}{x} \, dx,x,\sinh (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sinh ^n(x)\right )}{n}\\ &=\frac{2 \sqrt{a+b \sinh ^n(x)}}{n}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sinh ^n(x)\right )}{n}\\ &=\frac{2 \sqrt{a+b \sinh ^n(x)}}{n}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^n(x)}\right )}{b n}\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{n}+\frac{2 \sqrt{a+b \sinh ^n(x)}}{n}\\ \end{align*}
Mathematica [A] time = 0.0208877, size = 45, normalized size = 0.96 \[ \frac{2 \sqrt{a+b \sinh ^n(x)}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^n(x)}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 38, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( 2\,\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{n}}}{\sqrt{a}}} \right ) \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*} \int \sqrt{b \sinh \left (x\right )^{n} + a} \coth \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81846, size = 552, normalized size = 11.74 \begin{align*} \left [\frac{\sqrt{a} \log \left (\frac{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) - 2 \, \sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt{a} + 2 \, a}{\cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + \sinh \left (n \log \left (\sinh \left (x\right )\right )\right )}\right ) + 2 \, \sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a}}{n}, \frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a} \sqrt{-a}}{a}\right ) + \sqrt{b \cosh \left (n \log \left (\sinh \left (x\right )\right )\right ) + b \sinh \left (n \log \left (\sinh \left (x\right )\right )\right ) + a}\right )}}{n}\right ] \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 \sqrt{b \sinh \left (x\right )^{n} + a} \coth \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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