Optimal. Leaf size=47 \[ \frac {\sinh (c+d x)}{3 d (1+\cosh (c+d x))^2}+\frac {\sinh (c+d x)}{3 d (1+\cosh (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2729, 2727}
\begin {gather*} \frac {\sinh (c+d x)}{3 d (\cosh (c+d x)+1)}+\frac {\sinh (c+d x)}{3 d (\cosh (c+d x)+1)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2727
Rule 2729
Rubi steps
\begin {align*} \int \frac {1}{(1+\cosh (c+d x))^2} \, dx &=\frac {\sinh (c+d x)}{3 d (1+\cosh (c+d x))^2}+\frac {1}{3} \int \frac {1}{1+\cosh (c+d x)} \, dx\\ &=\frac {\sinh (c+d x)}{3 d (1+\cosh (c+d x))^2}+\frac {\sinh (c+d x)}{3 d (1+\cosh (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 34, normalized size = 0.72 \begin {gather*} \frac {4 \sinh (c+d x)+\sinh (2 (c+d x))}{6 d (1+\cosh (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.80, size = 30, normalized size = 0.64
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{d x +c}+1\right )}{3 d \left ({\mathrm e}^{d x +c}+1\right )^{3}}\) | \(26\) |
derivativedivides | \(\frac {-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{d}\) | \(30\) |
default | \(\frac {-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{d}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (43) = 86\).
time = 0.27, size = 90, normalized size = 1.91 \begin {gather*} \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} + 1\right )}} + \frac {2}{3 \, d {\left (3 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-3 \, d x - 3 \, c\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (43) = 86\).
time = 0.45, size = 113, normalized size = 2.40 \begin {gather*} -\frac {2 \, {\left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right ) + 3 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) + d\right )} \sinh \left (d x + c\right ) + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.50, size = 36, normalized size = 0.77 \begin {gather*} \begin {cases} - \frac {\tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 d} + \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (\cosh {\left (c \right )} + 1\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 25, normalized size = 0.53 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (d x + c\right )} + 1\right )}}{3 \, d {\left (e^{\left (d x + c\right )} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 25, normalized size = 0.53 \begin {gather*} -\frac {2\,\left (3\,{\mathrm {e}}^{c+d\,x}+1\right )}{3\,d\,{\left ({\mathrm {e}}^{c+d\,x}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________