Optimal. Leaf size=113 \[ \frac{23 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 \sqrt{7} d}-\frac{9 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{10 d} \]
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Rubi [A] time = 0.149902, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2793, 3023, 2752, 2662, 2654} \[ \frac{23 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 \sqrt{7} d}-\frac{9 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{10 d} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2752
Rule 2662
Rule 2654
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx &=-\frac{\sqrt{3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}-\frac{1}{10} \int \frac{3-6 \cos (c+d x)-6 \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)}{10 d}-\frac{\sqrt{3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}+\frac{1}{60} \int \frac{-6+54 \cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{10 d}-\frac{\sqrt{3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}-\frac{9}{40} \int \sqrt{3-4 \cos (c+d x)} \, dx+\frac{23}{40} \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{9 \sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{20 d}+\frac{23 F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{20 \sqrt{7} d}-\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{10 d}-\frac{\sqrt{3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 0.156989, size = 102, normalized size = 0.9 \[ \frac{-4 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (3 (c+d x))+23 \sqrt{4 \cos (c+d x)-3} F\left (\left .\frac{1}{2} (c+d x)\right |8\right )+9 \sqrt{4 \cos (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x)\right |8\right )}{20 d \sqrt{3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 3.38, size = 254, normalized size = 2.3 \begin{align*} -{\frac{1}{140\,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 ( -448\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +504\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +23\,\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 ) -63\,\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 ) -56\, \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 )^{3}}{\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 )^{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(-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 )^{3}}{\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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