Optimal. Leaf size=28 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
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Rubi [A] time = 0.0763304, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3230, 266, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 3230
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\coth (x)}{\sqrt{a+b \sinh ^3(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^3}} \, dx,x,\sinh (x)\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sinh ^3(x)\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^3(x)}\right )}{3 b}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right )}{3 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0235918, size = 28, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^3(x)}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.214, size = 21, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\it Artanh} \left ({\sqrt{a+b \left ( \sinh \left ( x \right ) \right ) ^{3}}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \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 \frac{\coth \left (x\right )}{\sqrt{b \sinh \left (x\right )^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\coth{\left (x \right )}}{\sqrt{a + b \sinh ^{3}{\left (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 \frac{\coth \left (x\right )}{\sqrt{b \sinh \left (x\right )^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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