Optimal. Leaf size=61 \[ \frac {2 \sqrt {2+3 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {3+2 \sec (c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 61, 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 = {3943, 2740}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2740
Rule 3943
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+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 = 0.04, size = 61, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {2+3 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {3+2 \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.27, size = 145, normalized size = 2.38
method | result | size |
default | \(\frac {\sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \EllipticF \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-\cos \left (d x +c \right )-2\right )}\) | \(145\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.30, size = 54, normalized size = 0.89 \begin {gather*} \frac {-i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )}{3 \, d} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\sqrt {2 \sec {\left (c + d x \right )} + 3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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