Optimal. Leaf size=14 \[ x+\frac {2 \sinh (x)}{1-\cosh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4477, 2759, 8}
\begin {gather*} x+\frac {2 \sinh (x)}{1-\cosh (x)} \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 [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 24, normalized size = 1.71 \begin {gather*} -2 \coth \left (\frac {x}{2}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2\left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.04, size = 26, normalized size = 1.86
method | result | size |
risch | \(x -\frac {4}{{\mathrm e}^{x}-1}\) | \(11\) |
default | \(-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 12, normalized size = 0.86 \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 = 0.35, size = 22, normalized size = 1.57 \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 = 0.71 \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 = 1.67, size = 10, normalized size = 0.71 \begin {gather*} x-\frac {4}{{\mathrm {e}}^x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________