Optimal. Leaf size=11 \[ \frac{\log (\tanh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0134658, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2620, 29} \[ \frac{\log (\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2620
Rule 29
Rubi steps
\begin{align*} \int \text{csch}(a+b x) \text{sech}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{\log (\tanh (a+b x))}{b}\\ \end{align*}
Mathematica [B] time = 0.0129567, size = 31, normalized size = 2.82 \[ 2 \left (\frac{\log (\sinh (a+b x))}{2 b}-\frac{\log (\cosh (a+b x))}{2 b}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \tanh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.63962, size = 68, normalized size = 6.18 \begin{align*} \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac{\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.28999, size = 154, normalized size = 14. \begin{align*} -\frac{\log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \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 \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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.19867, size = 47, normalized size = 4.27 \begin{align*} -\frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]