Optimal. Leaf size=62 \[ -\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]
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Rubi [A] time = 0.06, antiderivative size = 62, 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 = {2815} \[ -\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)-3}}{\sqrt {-\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]
Antiderivative was successfully verified.
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Rule 2815
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {-3+2 \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*}
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Mathematica [B] time = 0.69, size = 160, normalized size = 2.58 \[ \frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \cot (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\left ((2 \cos (c+d x)-3) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5} d (-\cos (c+d x))^{3/2} \sqrt {2 \cos (c+d x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 98, normalized size = 1.58 \[ \frac {2 \EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3+2 \cos \left (d x +c \right )}}{d \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\cos \left (d x +c \right )}} \]
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 \[ \int \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) - 3}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {2 \cos {\left (c + d x \right )} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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