Optimal. Leaf size=30 \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0153925, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2282, 50, 63, 203} \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \sqrt{-9+e^{2 t}} \, dt &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-9+t}}{t} \, dt,t,e^{2 t}\right )\\ &=\sqrt{-9+e^{2 t}}-\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9+t} t} \, dt,t,e^{2 t}\right )\\ &=\sqrt{-9+e^{2 t}}-9 \operatorname{Subst}\left (\int \frac{1}{9+t^2} \, dt,t,\sqrt{-9+e^{2 t}}\right )\\ &=\sqrt{-9+e^{2 t}}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{-9+e^{2 t}}\right )\\ \end{align*}
Mathematica [A] time = 0.0069408, size = 30, normalized size = 1. \[ \sqrt{e^{2 t}-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{e^{2 t}-9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 23, normalized size = 0.8 \begin{align*} -3\,\arctan \left ( 1/3\,\sqrt{-9+{{\rm e}^{2\,t}}} \right ) +\sqrt{-9+{{\rm e}^{2\,t}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.40053, size = 30, normalized size = 1. \begin{align*} \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21625, size = 72, normalized size = 2.4 \begin{align*} \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.1149, size = 22, normalized size = 0.73 \begin{align*} \begin{cases} \sqrt{e^{2 t} - 9} - 3 \operatorname{acos}{\left (3 e^{- t} \right )} & \text{for}\: e^{t} < \log{\left (3 \right )} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05762, size = 30, normalized size = 1. \begin{align*} \sqrt{e^{\left (2 \, t\right )} - 9} - 3 \, \arctan \left (\frac{1}{3} \, \sqrt{e^{\left (2 \, t\right )} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]