Optimal. Leaf size=13 \[ \tanh (x) \sqrt {a \cosh ^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 13, 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 = {3207, 2637} \[ \tanh (x) \sqrt {a \cosh ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {a \cosh ^2(x)} \, dx &=\left (\sqrt {a \cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\sqrt {a \cosh ^2(x)} \tanh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ \tanh (x) \sqrt {a \cosh ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.06, size = 69, normalized size = 5.31 \[ \frac {{\left (2 \, \cosh \relax (x) e^{x} \sinh \relax (x) + e^{x} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} - 1\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x) + \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 14, normalized size = 1.08 \[ -\frac {1}{2} \, \sqrt {a} {\left (e^{\left (-x\right )} - e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 15, normalized size = 1.15 \[ \frac {a \cosh \relax (x ) \sinh \relax (x )}{\sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 17, normalized size = 1.31 \[ -\frac {1}{2} \, \sqrt {a} e^{\left (-x\right )} + \frac {1}{2} \, \sqrt {a} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 17, normalized size = 1.31 \[ \sqrt {a}\,\mathrm {tanh}\relax (x)\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.45, size = 19, normalized size = 1.46 \[ \frac {\sqrt {a} \sqrt {\cosh ^{2}{\relax (x )}} \sinh {\relax (x )}}{\cosh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________