Optimal. Leaf size=11 \[ \frac{\log (\sinh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0059246, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3475} \[ \frac{\log (\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rubi steps
\begin{align*} \int \coth (a+b x) \, dx &=\frac{\log (\sinh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0094081, size = 19, normalized size = 1.73 \[ \frac{\log (\tanh (a+b x))+\log (\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.002, size = 30, normalized size = 2.7 \begin{align*} -{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02564, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (\sinh \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.44702, size = 88, normalized size = 8. \begin{align*} -\frac{b x - \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 [A] time = 0.440701, size = 27, normalized size = 2.45 \begin{align*} \begin{cases} x - \frac{\log{\left (\tanh{\left (a + b x \right )} + 1 \right )}}{b} + \frac{\log{\left (\tanh{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\x \coth{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18339, size = 38, normalized size = 3.45 \begin{align*} -\frac{b x + a}{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]