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 = 0.0385874, antiderivative size = 25, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0593641, size = 45, normalized size = 1.8 \[ \frac{2 \sqrt{4 \cos (c+d x)-3} \Pi \left (2;\left .\frac{1}{2} (c+d x)\right |8\right )}{d \sqrt{3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.503, size = 139, normalized size = 5.6 \begin{align*}{\frac{2}{7\,d}\sqrt{- \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticPi} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,2,{\frac{2\,\sqrt{14}}{7}} \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\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{\sec \left (d x + c\right )}{\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} \sec \left (d x + c\right )}{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{\sec{\left (c + d x \right )}}{\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{\sec \left (d x + c\right )}{\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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