Optimal. Leaf size=13 \[ \sqrt{-\cosh ^2(x)} \tanh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0256116, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3176, 3207, 2637} \[ \sqrt{-\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3176
Rule 3207
Rule 2637
Rubi steps
\begin{align*} \int \sqrt{-1-\sinh ^2(x)} \, dx &=\int \sqrt{-\cosh ^2(x)} \, dx\\ &=\left (\sqrt{-\cosh ^2(x)} \text{sech}(x)\right ) \int \cosh (x) \, dx\\ &=\sqrt{-\cosh ^2(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0053552, size = 13, normalized size = 1. \[ \sqrt{-\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 15, normalized size = 1.2 \begin{align*} -{\cosh \left ( x \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.55258, size = 34, normalized size = 2.62 \begin{align*} -\frac{e^{\left (-2 \, x\right )}}{2 \, \sqrt{-e^{\left (-2 \, x\right )}}} + \frac{1}{2 \, \sqrt{-e^{\left (-2 \, x\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.7685, size = 38, normalized size = 2.92 \begin{align*} \frac{1}{2} \,{\left (i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \sinh ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.30061, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2} i \, e^{\left (-x\right )} + \frac{1}{2} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]