Optimal. Leaf size=8 \[ 2 \coth (x)-x \]
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Rubi [A] time = 0.0433169, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3171, 3175, 3767, 8} \[ 2 \coth (x)-x \]
Antiderivative was successfully verified.
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Rule 3171
Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1+\cosh ^2(x)}{1-\cosh ^2(x)} \, dx &=-x+2 \int \frac{1}{1-\cosh ^2(x)} \, dx\\ &=-x-2 \int \text{csch}^2(x) \, dx\\ &=-x+2 i \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-x+2 \coth (x)\\ \end{align*}
Mathematica [A] time = 0.0061044, size = 8, normalized size = 1. \[ 2 \coth (x)-x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 28, normalized size = 3.5 \begin{align*} \tanh \left ({\frac{x}{2}} \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07999, size = 19, normalized size = 2.38 \begin{align*} -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 [B] time = 2.37461, size = 54, normalized size = 6.75 \begin{align*} -\frac{{\left (x + 2\right )} \sinh \left (x\right ) - 2 \, \cosh \left (x\right )}{\sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.58885, size = 12, normalized size = 1.5 \begin{align*} - x + \tanh{\left (\frac{x}{2} \right )} + \frac{1}{\tanh{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16469, size = 19, normalized size = 2.38 \begin{align*} -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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