Optimal. Leaf size=24 \[ \frac {2 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2662} \[ \frac {2 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]
Antiderivative was successfully verified.
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Rule 2662
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx &=\frac {2 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 1.83 \[ \frac {2 \sqrt {4 \cos (c+d x)-3} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{d \sqrt {3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 54, normalized size = 2.25 \[ \frac {2 \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \mathrm {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| 2 \sqrt {2}\right )}{d \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}} \]
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 \[ \int \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 39, normalized size = 1.62 \[ \frac {2\,\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )}{d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \]
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 \[ \int \frac {1}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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