Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{1+\cosh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4477, 2759, 8}
\begin {gather*} x-\frac {2 \sinh (x)}{\cosh (x)+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2759
Rule 4477
Rubi steps
\begin {align*} \int \frac {1}{(\coth (x)+\text {csch}(x))^2} \, dx &=-\int \frac {\sinh ^2(x)}{(i+i \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 = 0.02, size = 10, normalized size = 0.83 \begin {gather*} x-2 \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.99, size = 24, normalized size = 2.00
method | result | size |
risch | \(x +\frac {4}{{\mathrm e}^{x}+1}\) | \(11\) |
default | \(-2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 12, normalized size = 1.00 \begin {gather*} x - \frac {4}{e^{\left (-x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 20, normalized size = 1.67 \begin {gather*} \frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 4}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 10, normalized size = 0.83 \begin {gather*} x + \frac {4}{e^{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.18, size = 10, normalized size = 0.83 \begin {gather*} x+\frac {4}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________