Optimal. Leaf size=107 \[ -\frac{\sqrt{7} F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}+\frac{21 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}-\frac{\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{5 d} \]
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Rubi [A] time = 0.137994, antiderivative size = 107, 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 = {2791, 2753, 2752, 2662, 2654} \[ -\frac{\sqrt{7} F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}+\frac{21 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{20 d}-\frac{\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2791
Rule 2753
Rule 2752
Rule 2662
Rule 2654
Rubi steps
\begin{align*} \int \sqrt{3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx &=-\frac{(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac{1}{10} \int \sqrt{3-4 \cos (c+d x)} (-6-3 \cos (c+d x)) \, dx\\ &=\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac{(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac{1}{15} \int \frac{-21+\frac{63}{2} \cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac{(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}-\frac{7}{40} \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx+\frac{21}{40} \int \sqrt{3-4 \cos (c+d x)} \, dx\\ &=\frac{21 \sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{20 d}-\frac{\sqrt{7} F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{20 d}+\frac{\sqrt{3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac{(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 0.20503, size = 104, normalized size = 0.97 \[ -\frac{14 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))+7 \sqrt{4 \cos (c+d x)-3} F\left (\left .\frac{1}{2} (c+d x)\right |8\right )+21 \sqrt{4 \cos (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x)\right |8\right )}{20 d \sqrt{3-4 \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.527, size = 253, normalized size = 2.4 \begin{align*}{\frac{1}{20\,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 ( -256\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +128\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +\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 EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,{\frac{2\,\sqrt{14}}{7}} \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 ) -12\, \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 \sqrt{-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}\,{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 (\sqrt{-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}, 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 \sqrt{-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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