Optimal. Leaf size=24 \[ -\frac{1}{d (a \sinh (c+d x)+a \cosh (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0164745, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3071} \[ -\frac{1}{d (a \sinh (c+d x)+a \cosh (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3071
Rubi steps
\begin{align*} \int \frac{1}{a \cosh (c+d x)+a \sinh (c+d x)} \, dx &=-\frac{1}{d (a \cosh (c+d x)+a \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0382466, size = 24, normalized size = 1. \[ -\frac{1}{d (a \sinh (c+d x)+a \cosh (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 24, normalized size = 1. \begin{align*} -{\frac{1}{da \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08815, size = 23, normalized size = 0.96 \begin{align*} -\frac{e^{\left (-d x - c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9821, size = 59, normalized size = 2.46 \begin{align*} -\frac{1}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.624938, size = 34, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{1}{a d \sinh{\left (c + d x \right )} + a d \cosh{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{a \sinh{\left (c \right )} + a \cosh{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12213, size = 23, normalized size = 0.96 \begin{align*} -\frac{e^{\left (-d x - c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]