Optimal. Leaf size=18 \[ -\frac{1}{2 (\cosh (x)+1)}-\frac{1}{2} \tanh ^{-1}(\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0754002, antiderivative size = 24, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {4397, 2706, 2606, 30, 2611, 3770} \[ \frac{\text{csch}^2(x)}{2}-\frac{1}{2} \tanh ^{-1}(\cosh (x))-\frac{1}{2} \coth (x) \text{csch}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 2706
Rule 2606
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sinh (x)+\tanh (x)} \, dx &=-\left (i \int \frac{\coth (x)}{-i-i \cosh (x)} \, dx\right )\\ &=\int \coth ^2(x) \text{csch}(x) \, dx-\int \coth (x) \text{csch}^2(x) \, dx\\ &=-\frac{1}{2} \coth (x) \text{csch}(x)+\frac{1}{2} \int \text{csch}(x) \, dx-\operatorname{Subst}(\int x \, dx,x,-i \text{csch}(x))\\ &=-\frac{1}{2} \tanh ^{-1}(\cosh (x))-\frac{1}{2} \coth (x) \text{csch}(x)+\frac{\text{csch}^2(x)}{2}\\ \end{align*}
Mathematica [A] time = 0.026307, size = 35, normalized size = 1.94 \[ -\frac{1}{4} \text{sech}^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 17, normalized size = 0.9 \begin{align*}{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.13559, size = 53, normalized size = 2.94 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1} - \frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.81944, size = 386, normalized size = 21.44 \begin{align*} -\frac{{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh{\left (x \right )} + \tanh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15286, size = 58, normalized size = 3.22 \begin{align*} \frac{e^{\left (-x\right )} + e^{x} - 2}{4 \,{\left (e^{\left (-x\right )} + e^{x} + 2\right )}} - \frac{1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]