Optimal. Leaf size=23 \[ -\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2648} \[ -\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{1-\cosh (c+d x)} \, dx &=-\frac {\sinh (c+d x)}{d (1-\cosh (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 14, normalized size = 0.61 \[ \frac {\coth \left (\frac {1}{2} (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 24, normalized size = 1.04 \[ \frac {2}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 15, normalized size = 0.65 \[ \frac {2}{d {\left (e^{\left (d x + c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 16, normalized size = 0.70 \[ \frac {1}{d \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.39, size = 18, normalized size = 0.78 \[ -\frac {2}{d {\left (e^{\left (-d x - c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.89, size = 15, normalized size = 0.65 \[ \frac {2}{d\,\left ({\mathrm {e}}^{c+d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.63, size = 32, normalized size = 1.39 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\\frac {x}{1 - \cosh {\relax (c )}} & \text {for}\: d = 0 \\\frac {1}{d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________