Optimal. Leaf size=34 \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0604187, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2671} \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0641767, size = 34, normalized size = 1. \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 34, normalized size = 1. \begin{align*} -2\,{\frac{\cos \left ( dx+c \right ) }{d\sqrt{e\cos \left ( dx+c \right ) }\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58804, size = 176, normalized size = 5.18 \begin{align*} -\frac{2 \,{\left (\sqrt{a} \sqrt{e} - \frac{\sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{{\left (a e + \frac{a e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}} \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43498, size = 107, normalized size = 3.15 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{a d e \sin \left (d x + c\right ) + a d e} \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{a \left (\sin{\left (c + d x \right )} + 1\right )} \sqrt{e \cos{\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{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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