Optimal. Leaf size=54 \[ -\frac{2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{b \sqrt{\sinh (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.019737, antiderivative size = 54, 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 = {2642, 2641} \[ -\frac{2 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx &=\frac{\sqrt{i \sinh (a+b x)} \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{\sqrt{\sinh (a+b x)}}\\ &=-\frac{2 i F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{b \sqrt{\sinh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.114061, size = 48, normalized size = 0.89 \[ -\frac{2 \sqrt{\sinh (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 i a-2 i b x+\pi ),2\right )}{b \sqrt{i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 87, normalized size = 1.6 \begin{align*}{\frac{i\sqrt{2}}{b\cosh \left ( bx+a \right ) }\sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) }\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \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{\sinh \left (b x + a\right )}}\,{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{1}{\sqrt{\sinh \left (b x + a\right )}}, 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{\sinh{\left (a + b x \right )}}}\, 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{\sinh \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]