Optimal. Leaf size=8 \[ 2 \coth (x)-x \]
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Rubi [A] time = 0.04, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 8
Rule 3171
Rule 3175
Rule 3767
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*}
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Mathematica [A] time = 0.01, size = 8, normalized size = 1.00 \[ 2 \coth (x)-x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 17, normalized size = 2.12 \[ -\frac {{\left (x + 2\right )} \sinh \relax (x) - 2 \, \cosh \relax (x)}{\sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 14, normalized size = 1.75 \[ -x + \frac {4}{e^{\left (2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 28, normalized size = 3.50 \[ \tanh \left (\frac {x}{2}\right )+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 14, normalized size = 1.75 \[ -x - \frac {4}{e^{\left (-2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 14, normalized size = 1.75 \[ \frac {4}{{\mathrm {e}}^{2\,x}-1}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.91, size = 12, normalized size = 1.50 \[ - x + \tanh {\left (\frac {x}{2} \right )} + \frac {1}{\tanh {\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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