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 = 0.0119922, antiderivative size = 24, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.034354, 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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Maple [C] time = 0.163, size = 54, normalized size = 2.3 \begin{align*} 2\,{\frac{\sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it InverseJacobiAM} \left ( 1/2\,dx+c/2,2\,\sqrt{2} \right ) }{d\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 - 4 \cos{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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