Optimal. Leaf size=99 \[ -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right )}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 99, 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, 2809} \[ -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right )}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2810
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}}\\ &=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.77, size = 145, normalized size = 1.46 \[ -\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {-\frac {(2-3 \cos (c+d x))^2}{(\cos (c+d x)+1)^2}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \left (F\left (\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |5\right )-2 \Pi \left (-1;\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |5\right )\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {\frac {2-3 \cos (c+d x)}{\cos (c+d x)+1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right ) - 2}, x\right ) \]
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 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 144, normalized size = 1.45 \[ \frac {2 \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \sqrt {5}\right )\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+2\right ) \sqrt {\cos \left (d x +c \right )}} \]
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 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {2 - 3 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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