Optimal. Leaf size=47 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )\right |5\right )}{d \sqrt {-\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2814, 2813} \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )\right |5\right )}{d \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2813
Rule 2814
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {-2+3 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )\right |5\right )}{d \sqrt {-\cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.40, size = 158, normalized size = 3.36 \[ \frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\left ((3 \cos (c+d x)-2) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\left ((3 \cos (c+d x)-2) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )}\right )|\frac {4}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)} \sqrt {3 \cos (c+d x)-2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) - 2}}{3 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.19, size = 109, normalized size = 2.32 \[ \frac {2 \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {-\cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {3\,\cos \left (c+d\,x\right )-2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {3 \cos {\left (c + d x \right )} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________