Optimal. Leaf size=45 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right ),\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e} \]
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Rubi [A] time = 0.0372017, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3126, 2661} \[ \frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e} \]
Antiderivative was successfully verified.
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Rule 3126
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx &=\int \frac{1}{\sqrt{2+\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )}} \, dx\\ &=\frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{\sqrt{2+\sqrt{34}} e}\\ \end{align*}
Mathematica [C] time = 0.261814, size = 128, normalized size = 2.84 \[ \frac{\sqrt{\frac{2}{15}} \sqrt{\sqrt{34} \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+2} \sqrt{\cos ^2\left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )} \sec \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right ) F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )}{e} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.134, size = 158, normalized size = 3.5 \begin{align*}{\frac{ \left ( 2\,\sqrt{34}-34 \right ) \sqrt{17}}{17\,\cos \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) e}\sqrt{-{\frac{17\,\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) +\sqrt{34}}{-\sqrt{34}+17}}}\sqrt{-17\,{\frac{\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) -1}{\sqrt{34}+17}}}\sqrt{{\frac{1+\sin \left ( ex+d+\arctan \left ({\frac{3}{5}} \right ) \right ) }{-\sqrt{34}+17}}}{\it EllipticF} \left ( \sqrt{-{\frac{17\,\sin \left ( ex+d+\arctan \left ( 3/5 \right ) \right ) +\sqrt{34}}{-\sqrt{34}+17}}},i\sqrt{{\frac{-\sqrt{34}+17}{\sqrt{34}+17}}} \right ){\frac{1}{\sqrt{\sqrt{34}\sin \left ( ex+d+\arctan \left ({\frac{3}{5}} \right ) \right ) +2}}}} \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{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}\,{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{1}{\sqrt{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}, 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{5 \sin{\left (d + e x \right )} + 3 \cos{\left (d + e x \right )} + 2}}\, 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{3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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