Optimal. Leaf size=80 \[ \frac{17 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{4 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{6 d} \]
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Rubi [A] time = 0.10212, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2791, 2752, 2662, 2654} \[ \frac{17 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{4 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{6 d} \]
Antiderivative was successfully verified.
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Rule 2791
Rule 2752
Rule 2662
Rule 2654
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx &=-\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{6 d}-\frac{1}{6} \int \frac{-2-3 \cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{6 d}-\frac{1}{8} \int \sqrt{3-4 \cos (c+d x)} \, dx+\frac{17}{24} \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 )}{4 d}+\frac{17 F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{12 \sqrt{7} d}-\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.116625, size = 94, normalized size = 1.18 \[ \frac{-6 \sin (c+d x)+4 \sin (2 (c+d x))+17 \sqrt{4 \cos (c+d x)-3} F\left (\left .\frac{1}{2} (c+d x)\right |8\right )+3 \sqrt{4 \cos (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x)\right |8\right )}{12 d \sqrt{3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.379, size = 232, normalized size = 2.9 \begin{align*} -{\frac{1}{84\,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}} \left ( 224\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +17\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) -21\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) -28\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \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 )^{2}}{\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 )^{2}}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{2}}{\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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