Optimal. Leaf size=122 \[ \frac {\sqrt {5} \sqrt {3 \sec (c+d x)+2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}-\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{\sqrt {5} d \sqrt {3 \sec (c+d x)+2}} \]
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Rubi [A] time = 0.18, antiderivative size = 122, 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} \[ \frac {\sqrt {5} \sqrt {3 \sec (c+d x)+2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}-\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{\sqrt {5} d \sqrt {3 \sec (c+d x)+2}} \]
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 {\sec (c+d x)} \sqrt {2+3 \sec (c+d x)}} \, dx &=\frac {1}{2} \int \frac {\sqrt {2+3 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx-\frac {3}{2} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {2+3 \sec (c+d x)}} \, dx\\ &=-\frac {\left (3 \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {3+2 \cos (c+d x)}} \, dx}{2 \sqrt {2+3 \sec (c+d x)}}+\frac {\sqrt {2+3 \sec (c+d x)} \int \sqrt {3+2 \cos (c+d x)} \, dx}{2 \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=-\frac {3 \sqrt {3+2 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {2+3 \sec (c+d x)}}+\frac {\sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {2+3 \sec (c+d x)}}{d \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 78, normalized size = 0.64 \[ \frac {\sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )-3 F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )\right )}{\sqrt {5} d \sqrt {3 \sec (c+d x)+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, \sec \left (d x + c\right ) + 2} \sqrt {\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right )^{2} + 2 \, \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, \sec \left (d x + c\right ) + 2} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.77, size = 405, normalized size = 3.32 \[ -\frac {\left (2 i \sin \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sqrt {5}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {2}+i \sin \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sqrt {5}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {2}+2 i \sqrt {5}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+i \sqrt {5}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+20 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right )-30\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{10 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (3+2 \cos \left (d x +c \right )\right )} \]
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 \[ \int \frac {1}{\sqrt {3 \, \sec \left (d x + c\right ) + 2} \sqrt {\sec \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {3}{\cos \left (c+d\,x\right )}+2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 \sec {\left (c + d x \right )} + 2} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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