Optimal. Leaf size=44 \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}} \]
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Rubi [A] time = 0.0173385, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3112} \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}} \]
Antiderivative was successfully verified.
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Rule 3112
Rubi steps
\begin{align*} \int \sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx &=-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)}}\\ \end{align*}
Mathematica [A] time = 0.0417613, size = 75, normalized size = 1.7 \[ \frac{2 \left (\sin \left (\frac{1}{2} (d+e x)\right )+3 \cos \left (\frac{1}{2} (d+e x)\right )\right ) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}{e \left (\cos \left (\frac{1}{2} (d+e x)\right )-3 \sin \left (\frac{1}{2} (d+e x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.077, size = 50, normalized size = 1.1 \begin{align*} 10\,{\frac{ \left ( \sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) -1 \right ) \left ( 1+\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \right ) }{\cos \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \sqrt{-5+5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) }e}} \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{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76369, size = 163, normalized size = 3.7 \begin{align*} \frac{2 \, \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}{\left (3 \, \cos \left (e x + d\right ) + \sin \left (e x + d\right ) + 3\right )}}{e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + 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 \sqrt{3 \sin{\left (d + e x \right )} + 4 \cos{\left (d + e x \right )} - 5}\, 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{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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