Optimal. Leaf size=13 \[ \tanh (x) \sqrt{a \cosh ^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0122538, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 2637} \[ \tanh (x) \sqrt{a \cosh ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3207
Rule 2637
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 = 0.0043494, size = 13, normalized size = 1. \[ \tanh (x) \sqrt{a \cosh ^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 15, normalized size = 1.2 \begin{align*}{a\cosh \left ( x \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59789, size = 23, normalized size = 1.77 \begin{align*} -\frac{1}{2} \, \sqrt{a} e^{\left (-x\right )} + \frac{1}{2} \, \sqrt{a} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.39978, size = 216, normalized size = 16.62 \begin{align*} \frac{{\left (2 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + e^{x} \sinh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{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 \left (x\right ) e^{\left (2 \, x\right )} +{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.617942, size = 19, normalized size = 1.46 \begin{align*} \frac{\sqrt{a} \sqrt{\cosh ^{2}{\left (x \right )}} \sinh{\left (x \right )}}{\cosh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15647, size = 19, normalized size = 1.46 \begin{align*} -\frac{1}{2} \, \sqrt{a}{\left (e^{\left (-x\right )} - e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]