Optimal. Leaf size=77 \[ -\frac {4 \cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right )}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2809} \[ -\frac {4 \cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 2809
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx &=-\frac {4 \cot (c+d x) \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {2+3 \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)}}{3 d}\\ \end {align*}
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Mathematica [B] time = 2.84, size = 175, normalized size = 2.27 \[ \frac {2 \sqrt {\cos (c+d x)} \sqrt {3 \cos (c+d x)+2} \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \left (3 F\left (\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )-5 \Pi \left (-\frac {2}{3};\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )\right )}{3 d \sqrt {\frac {-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)-1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) + 2}}, x\right ) \]
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 [B] time = 0.21, size = 142, normalized size = 1.84 \[ -\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {\sqrt {5}}{5}\right )\right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}} \]
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 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) + 2}}\,{d x} \]
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 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \]
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 \[ \int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {3 \cos {\left (c + d x \right )} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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