Optimal. Leaf size=61 \[ \frac {2 \sqrt {3+2 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {-2-3 \sec (c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3943, 2742,
2740} \begin {gather*} \frac {2 \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2740
Rule 2742
Rule 3943
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx &=\frac {\left (\sqrt {-3-2 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {-3-2 \cos (c+d x)}} \, dx}{\sqrt {-2-3 \sec (c+d x)}}\\ &=\frac {\left (\sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {3}{5}+\frac {2}{5} \cos (c+d x)}} \, dx}{\sqrt {5} \sqrt {-2-3 \sec (c+d x)}}\\ &=\frac {2 \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)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 1.00 \begin {gather*} \frac {2 \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)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.22, size = 139, normalized size = 2.28
method | result | size |
default | \(\frac {i \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sqrt {-\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {\sqrt {5}}{5}\right )}{5 d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right )-3\right )}\) | \(139\) |
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 = 0.58, size = 44, normalized size = 0.72 \begin {gather*} -\frac {{\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) + {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right )}{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 {- 3 \sec {\left (c + d x \right )} - 2}}\, 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 {3}{\cos \left (c+d\,x\right )}-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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