Optimal. Leaf size=22 \[ -\frac {1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]
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Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {4385} \[ -\frac {1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin {align*} \int \frac {\cosh (a+b x)-\sinh (a+b x)}{\cosh (a+b x)+\sinh (a+b x)} \, dx &=-\frac {1}{2 b (\cosh (a+b x)+\sinh (a+b x))^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 65, normalized size = 2.95 \[ -\frac {\sinh (2 a) \sinh (2 b x)}{2 b}-\frac {\cosh (2 a) \cosh (2 b x)}{2 b}+\frac {\sinh (2 a) \cosh (2 b x)}{2 b}+\frac {\cosh (2 a) \sinh (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 40, normalized size = 1.82 \[ -\frac {1}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 14, normalized size = 0.64 \[ -\frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 1.64 \[ -\frac {\cosh \left (b x +a \right )-\sinh \left (b x +a \right )}{2 b \left (\cosh \left (b x +a \right )+\sinh \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 14, normalized size = 0.64 \[ -\frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 14, normalized size = 0.64 \[ -\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 37, normalized size = 1.68 \[ \begin {cases} \frac {\sinh {\left (a + b x \right )}}{b \sinh {\left (a + b x \right )} + b \cosh {\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \left (- \sinh {\relax (a )} + \cosh {\relax (a )}\right )}{\sinh {\relax (a )} + \cosh {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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