Optimal. Leaf size=53 \[ \frac{3 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{2 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{2 d} \]
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Rubi [A] time = 0.0501004, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2752, 2662, 2654} \[ \frac{3 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{2 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2752
Rule 2662
Rule 2654
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx &=-\left (\frac{1}{4} \int \sqrt{3-4 \cos (c+d x)} \, dx\right )+\frac{3}{4} \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{2 d}+\frac{3 F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{2 \sqrt{7} d}\\ \end{align*}
Mathematica [A] time = 0.0667908, size = 60, normalized size = 1.13 \[ \frac{\sqrt{4 \cos (c+d x)-3} \left (3 F\left (\left .\frac{1}{2} (c+d x)\right |8\right )+E\left (\left .\frac{1}{2} (c+d x)\right |8\right )\right )}{2 d \sqrt{3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.565, size = 158, normalized size = 3. \begin{align*} -{\frac{1}{14\,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} \left ( 3\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) -7\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) \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{\cos \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} \cos \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{\cos{\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{\cos \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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