Optimal. Leaf size=27 \[ -\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2813} \[ -\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2813
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx &=-\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right )\right |5\right )}{d}\\ \end {align*}
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Mathematica [B] time = 0.49, size = 155, normalized size = 5.74 \[ \frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)-1}}\right )|\frac {4}{5}\right )}{\sqrt {5} d \sqrt {-3 \cos (c+d x)-2} \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, 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 = 0.18, size = 129, normalized size = 4.78 \[ -\frac {\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right ) \sqrt {-\cos \left (d x +c \right )}} \]
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 = -0.04 \[ \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \]
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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