Optimal. Leaf size=13 \[ -\frac {x}{a}+\frac {\sinh (x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2761, 8}
\begin {gather*} \frac {\sinh (x)}{a}-\frac {x}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2761
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{a+a \cosh (x)} \, dx &=\frac {\sinh (x)}{a}-\frac {\int 1 \, dx}{a}\\ &=-\frac {x}{a}+\frac {\sinh (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 17, normalized size = 1.31 \begin {gather*} \frac {2 \left (-\frac {x}{2}+\frac {\sinh (x)}{2}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs.
\(2(13)=26\).
time = 0.48, size = 45, normalized size = 3.46
method | result | size |
risch | \(-\frac {x}{a}+\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}\) | \(24\) |
default | \(\frac {-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 23, normalized size = 1.77 \begin {gather*} -\frac {x}{a} - \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.42, size = 11, normalized size = 0.85 \begin {gather*} -\frac {x - \sinh \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (7) = 14\).
time = 0.17, size = 46, normalized size = 3.54 \begin {gather*} - \frac {x \tanh ^{2}{\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} + \frac {x}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} - \frac {2 \tanh {\left (\frac {x}{2} \right )}}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 17, normalized size = 1.31 \begin {gather*} -\frac {2 \, x + e^{\left (-x\right )} - e^{x}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.91, size = 23, normalized size = 1.77 \begin {gather*} \frac {{\mathrm {e}}^x}{2\,a}-\frac {x}{a}-\frac {{\mathrm {e}}^{-x}}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________