Optimal. Leaf size=27 \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3077, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 3077
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}} \, dx &=\frac {\int \frac {1}{\sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}} \, dx}{\sqrt [4]{13}}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{\sqrt [4]{13} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.09, size = 88, normalized size = 3.26 \[ \frac {2 \sqrt {-\left (\left (\sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )-1\right ) \sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )} \sqrt {\sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )+1} \sec \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )}{\sqrt [4]{13} d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 85, normalized size = 3.15 \[ \frac {\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \sqrt {\sqrt {13}\, \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \sin {\left (c + d x \right )} + 2 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________