Optimal. Leaf size=97 \[ -\frac {3 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {5}{2};\left .\sin ^{-1}\left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt {-\cos (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2810, 2808} \[ -\frac {3 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {5}{2};\left .\sin ^{-1}\left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2808
Rule 2810
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}}\\ &=-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \Pi \left (\frac {5}{2};\left .\sin ^{-1}\left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d \sqrt {-\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 113, normalized size = 1.16 \[ -\frac {2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x) (2 \cos (c+d x)+3) \sec ^4\left (\frac {1}{2} (c+d x)\right )} \left (F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )-2 \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-2 \cos (c+d x)-3} \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, \cos \left (d x + c\right ) - 3} \sqrt {\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right ) + 3}, x\right ) \]
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 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 168, normalized size = 1.73 \[ -\frac {i \sqrt {2}\, \sqrt {10}\, \left (\EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right )-2 \EllipticPi \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, 5, i \sqrt {5}\right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3-2 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {5}}{5 d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right )-3\right ) \sqrt {\cos \left (d x +c \right )}} \]
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 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {-2\,\cos \left (c+d\,x\right )-3}} \,d x \]
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 \[ \int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {- 2 \cos {\left (c + d x \right )} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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