Optimal. Leaf size=14 \[ x+\frac{2 \sinh (x)}{1-\cosh (x)} \]
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Rubi [A] time = 0.0822005, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4392, 2670, 2680, 8} \[ x+\frac{2 \sinh (x)}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int (\coth (x)+\text{csch}(x))^2 \, dx &=-\int (i+i \cosh (x))^2 \text{csch}^2(x) \, dx\\ &=-\int \frac{\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac{2 \sinh (x)}{1-\cosh (x)}+\int 1 \, dx\\ &=x+\frac{2 \sinh (x)}{1-\cosh (x)}\\ \end{align*}
Mathematica [A] time = 0.0258454, size = 10, normalized size = 0.71 \[ x-2 \coth \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 21, normalized size = 1.5 \begin{align*} x-2\,{\rm coth} \left (x\right )-2\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\sinh \left ( x \right ) }}+2\,\sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04969, size = 34, normalized size = 2.43 \begin{align*} x + \frac{4}{e^{\left (-x\right )} - e^{x}} + \frac{4}{e^{\left (-2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96929, size = 77, normalized size = 5.5 \begin{align*} \frac{x \cosh \left (x\right ) + x \sinh \left (x\right ) - x - 4}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1} \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 \left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10915, size = 14, normalized size = 1. \begin{align*} x - \frac{4}{e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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