Optimal. Leaf size=62 \[ \frac {2 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{\sqrt {5} d \sqrt {2 \sec (c+d x)-3}} \]
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Rubi [A] time = 0.06, antiderivative size = 62, 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 = {3858, 2662} \[ \frac {2 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{\sqrt {5} d \sqrt {2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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Rule 2662
Rule 3858
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3+2 \sec (c+d x)}} \, dx &=\frac {\left (\sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {2-3 \cos (c+d x)}} \, dx}{\sqrt {-3+2 \sec (c+d x)}}\\ &=\frac {2 \sqrt {2-3 \cos (c+d x)} F\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 0.87 \[ \frac {2 \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{d \sqrt {2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {2 \, \sec \left (d x + c\right ) - 3}}, x\right ) \]
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 \[ \int \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {2 \, \sec \left (d x + c\right ) - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.61, size = 136, normalized size = 2.19 \[ \frac {2 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {-\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+2\right )} \]
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 \[ \int \frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {2 \, \sec \left (d x + c\right ) - 3}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}-3}} \,d x \]
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 \[ \int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {2 \sec {\left (c + d x \right )} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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