Optimal. Leaf size=55 \[ \frac {2 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{d \sqrt {-2 \sec (c+d x)-3}} \]
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Rubi [A] time = 0.06, antiderivative size = 55, 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} \[ \frac {2 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{d \sqrt {-2 \sec (c+d x)-3}} \]
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 (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {\sec (c+d x)}}{d \sqrt {-3-2 \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 1.11 \[ \frac {2 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{\sqrt {5} d \sqrt {-2 \sec (c+d x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right ) + 3}, x\right ) \]
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 = 1.68, size = 142, normalized size = 2.58 \[ \frac {i \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {-\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right )} \]
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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