Optimal. Leaf size=113 \[ \frac {4 \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{3 d \sqrt {3-2 \sec (c+d x)}}+\frac {2 \sqrt {3-2 \sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{3 d \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.18, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3862, 3856, 2653, 3858, 2661} \[ \frac {4 \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{3 d \sqrt {3-2 \sec (c+d x)}}+\frac {2 \sqrt {3-2 \sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |6\right )}{3 d \sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3856
Rule 3858
Rule 3862
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx &=\frac {1}{3} \int \frac {\sqrt {3-2 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx+\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {3-2 \sec (c+d x)}} \, dx\\ &=\frac {\sqrt {3-2 \sec (c+d x)} \int \sqrt {-2+3 \cos (c+d x)} \, dx}{3 \sqrt {-2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {\left (2 \sqrt {-2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {-2+3 \cos (c+d x)}} \, dx}{3 \sqrt {3-2 \sec (c+d x)}}\\ &=\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |6\right ) \sqrt {3-2 \sec (c+d x)}}{3 d \sqrt {-2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {4 \sqrt {-2+3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |6\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {3-2 \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 72, normalized size = 0.64 \[ \frac {\sqrt {3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} \left (4 F\left (\left .\frac {1}{2} (c+d x)\right |6\right )+2 E\left (\left .\frac {1}{2} (c+d x)\right |6\right )\right )}{3 d \sqrt {3-2 \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right )^{2} - 3 \, \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.91, size = 381, normalized size = 3.37 \[ \frac {2 \left (3 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}-5 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {5}+3 \sqrt {5}\, \EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-5 \sqrt {5}\, \EllipticE \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-15 \left (\cos ^{2}\left (d x +c \right )\right )+25 \cos \left (d x +c \right )-10\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{15 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (-2+3 \cos \left (d x +c \right )\right )} \]
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 \[ \int \frac {1}{\sqrt {-2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {3-\frac {2}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 - 2 \sec {\left (c + d x \right )}} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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