Optimal. Leaf size=56 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]
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Rubi [A] time = 0.0215649, 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 = {2640, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{b \sinh (c+d x)} \, dx &=\frac{\sqrt{b \sinh (c+d x)} \int \sqrt{i \sinh (c+d x)} \, dx}{\sqrt{i \sinh (c+d x)}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0450161, size = 52, normalized size = 0.93 \[ \frac{2 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right ) \sqrt{b \sinh (c+d x)}}{d \sqrt{i \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 111, normalized size = 2. \begin{align*}{\frac{b\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 ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) \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 \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 (\sqrt{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 \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 \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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