Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2680, 8} \[ x-\frac {2 \sinh (x)}{\cosh (x)+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2680
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx &=-\frac {2 \sinh (x)}{1+\cosh (x)}+\int 1 \, dx\\ &=x-\frac {2 \sinh (x)}{1+\cosh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 1.50 \[ 2 \tanh ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.08, size = 20, normalized size = 1.67 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + x + 4}{\cosh \relax (x) + \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 10, normalized size = 0.83 \[ x + \frac {4}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 24, normalized size = 2.00 \[ -2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 12, normalized size = 1.00 \[ x - \frac {4}{e^{\left (-x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.90, size = 10, normalized size = 0.83 \[ x+\frac {4}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.55, size = 7, normalized size = 0.58 \[ x - 2 \tanh {\left (\frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________