Optimal. Leaf size=60 \[ \frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2815} \[ \frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2815
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx &=\frac {2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.04, size = 144, normalized size = 2.40 \[ \frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {(3-2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {\cos (c+d x)}{\cos (c+d x)-1}}}{\sqrt {3}}\right )\right |6\right )}{d \sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.19, size = 125, normalized size = 2.08 \[ \frac {\sqrt {2}\, \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{\frac {3}{2}} \left (-3+2 \cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {3-2\,\cos \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 - 2 \cos {\left (c + d x \right )}} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________