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 = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 1.70 \[ -\frac {2 \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}{e \left (\sin \left (\frac {1}{2} (d+e x)\right )+3 \cos \left (\frac {1}{2} (d+e x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.84, size = 61, normalized size = 1.39 \[ -\frac {2 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}{3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 50, normalized size = 1.14 \[ \frac {10 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right )}{\cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 39, normalized size = 0.89 \[ -\frac {2\,\sqrt {5}\,\left (3\,\cos \left (d+e\,x\right )-4\,\sin \left (d+e\,x\right )\right )}{5\,e\,\sqrt {\cos \left (d-\mathrm {atan}\left (\frac {3}{4}\right )+e\,x\right )+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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