Optimal. Leaf size=14 \[ \frac{1}{b (\tanh (a+b x)+1)} \]
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Rubi [A] time = 0.206698, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {32} \[ \frac{1}{b (\tanh (a+b x)+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin{align*} \int \frac{-\text{csch}(a+b x)+\text{sech}(a+b x)}{\text{csch}(a+b x)+\text{sech}(a+b x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{1}{b (1+\tanh (a+b x))}\\ \end{align*}
Mathematica [B] time = 0.0248082, size = 65, normalized size = 4.64 \[ \frac{\sinh (2 a) \sinh (2 b x)}{2 b}+\frac{\cosh (2 a) \cosh (2 b x)}{2 b}-\frac{\sinh (2 a) \cosh (2 b x)}{2 b}-\frac{\cosh (2 a) \sinh (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.129, size = 36, normalized size = 2.6 \begin{align*}{\frac{1}{b} \left ( 2\, \left ( \tanh \left ( 1/2\,bx+a/2 \right ) +1 \right ) ^{-2}-2\, \left ( \tanh \left ( 1/2\,bx+a/2 \right ) +1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10967, size = 19, normalized size = 1.36 \begin{align*} \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0868, size = 107, normalized size = 7.64 \begin{align*} \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 )}} \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{\operatorname{csch}{\left (a + b x \right )}}{\operatorname{csch}{\left (a + b x \right )} + \operatorname{sech}{\left (a + b x \right )}}\, dx - \int - \frac{\operatorname{sech}{\left (a + b x \right )}}{\operatorname{csch}{\left (a + b x \right )} + \operatorname{sech}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13312, size = 19, normalized size = 1.36 \begin{align*} \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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