Optimal. Leaf size=25 \[ -\frac {2 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2806} \[ -\frac {2 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
Antiderivative was successfully verified.
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Rule 2806
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx &=-\frac {2 \Pi \left (2;\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 45, normalized size = 1.80 \[ \frac {2 \sqrt {4 \cos (c+d x)-3} \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )}{d \sqrt {3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 139, normalized size = 5.56 \[ \frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]
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 \[ \int \frac {\sec \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\cos \left (c+d\,x\right )\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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