Optimal. Leaf size=56 \[ -\frac{2 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right )}{d \sqrt{b \sinh (c+d x)}} \]
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Rubi [A] time = 0.0218033, antiderivative size = 56, 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 = {2642, 2641} \[ -\frac{2 i \sqrt{i \sinh (c+d x)} F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{b \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \sinh (c+d x)}} \, dx &=\frac{\sqrt{i \sinh (c+d x)} \int \frac{1}{\sqrt{i \sinh (c+d x)}} \, dx}{\sqrt{b \sinh (c+d x)}}\\ &=-\frac{2 i F\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{i \sinh (c+d x)}}{d \sqrt{b \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0340537, size = 54, normalized size = 0.96 \[ \frac{2 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (\frac{\pi }{2}-i (c+d x)\right ),2\right )}{d \sqrt{b \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 89, normalized size = 1.6 \begin{align*}{\frac{i\sqrt{2}}{d\cosh \left ( dx+c \right ) }\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b\sinh \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sinh \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sinh \left (d x + c\right )}}{b \sinh \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sinh{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sinh \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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