Optimal. Leaf size=45 \[ \frac{2 \sqrt{2+\sqrt{34}} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0305556, 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 = {3118, 2653} \[ \frac{2 \sqrt{2+\sqrt{34}} E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )|\frac{2}{15} \left (17-\sqrt{34}\right )\right )}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3118
Rule 2653
Rubi steps
\begin{align*} \int \sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx &=\int \sqrt{2+\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )} \, dx\\ &=\frac{2 \sqrt{2+\sqrt{34}} E\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 )}{e}\\ \end{align*}
Mathematica [C] time = 2.30288, size = 326, normalized size = 7.24 \[ \frac{\sqrt{\sin ^2\left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )} \left (2 \sqrt{30} \sqrt{\sqrt{34} \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+2} \sqrt{\cos ^2\left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )} \sqrt{\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+2} \sec \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right ) F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left (d+e x+\tan ^{-1}\left (\frac{3}{5}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )+45 \sin (d+e x)-75 \cos (d+e x)\right )-15 \sqrt{30} \sin \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right ) F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+\sqrt{34}}{17+\sqrt{34}}\right )}{15 e \sqrt{\sin ^2\left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )} \sqrt{\sqrt{34} \cos \left (d+e x-\tan ^{-1}\left (\frac{5}{3}\right )\right )+2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 2.867, size = 316, normalized size = 7. \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}}} \left ({\it EllipticE} \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 ) \sqrt{34}-\sqrt{34}{\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 ) +2\,{\it EllipticE} \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 ) -2\,{\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 ) \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.
[In]
[Out]