Optimal. Leaf size=54 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 54, 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} \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\sin ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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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 (\sin ^{-1}\left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right )|\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 0.61, size = 150, normalized size = 2.78 \[ -\frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )}{d \sqrt {-\cos (c+d x)} \sqrt {3 \cos (c+d x)+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, 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.
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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.
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maple [B] time = 0.19, size = 122, normalized size = 2.26 \[ \frac {\EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {-\cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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