Optimal. Leaf size=80 \[ \frac {2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt {-\cos (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2817, 2815} \[ \frac {2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt {-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2815
Rule 2817
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}}\\ &=\frac {2 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |-5\right ) \sqrt {-\tan ^2(c+d x)}}{d \sqrt {-\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 154, normalized size = 1.92 \[ -\frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2 \cos (c+d x)+3) \csc ^2\left (\frac {1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {(2 \cos (c+d x)+3) \csc ^2\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {6}}\right )\right |6\right )}{d \sqrt {-\cos (c+d x)} \sqrt {2 \cos (c+d x)+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, 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 = 0.16, size = 127, normalized size = 1.59 \[ -\frac {i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {-\cos \left (d x +c \right )}} \]
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 \[ \int \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \]
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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