Optimal. Leaf size=39 \[ -\frac{i \sqrt{\cosh ^2(x)+1} \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right )}{\sqrt{-\cosh ^2(x)-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0202647, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3183, 3182} \[ -\frac{i \sqrt{\cosh ^2(x)+1} F\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{\sqrt{-\cosh ^2(x)-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\cosh ^2(x)}} \, dx &=\frac{\sqrt{1+\cosh ^2(x)} \int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx}{\sqrt{-1-\cosh ^2(x)}}\\ &=-\frac{i \sqrt{1+\cosh ^2(x)} F\left (\left .\frac{\pi }{2}+i x\right |-1\right )}{\sqrt{-1-\cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0439794, size = 40, normalized size = 1.03 \[ -\frac{i \sqrt{\cosh (2 x)+3} \text{EllipticF}\left (i x,\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\cosh (2 x)-3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.224, size = 61, normalized size = 1.6 \begin{align*}{\frac{{\it EllipticF} \left ( \cosh \left ( x \right ) ,i \right ) }{\sinh \left ( x \right ) }\sqrt{- \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1- \left ( \cosh \left ( x \right ) \right ) ^{4}}}}{\frac{1}{\sqrt{-1- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cosh \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{2 \, \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1}, 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 \frac{1}{\sqrt{- \cosh ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cosh \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]