Optimal. Leaf size=23 \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]
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Rubi [A] time = 0.011538, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2653} \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2653
Rubi steps
\begin{align*} \int \sqrt{3+4 \cos (c+d x)} \, dx &=\frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0236297, size = 23, normalized size = 1. \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.635, size = 137, normalized size = 6. \begin{align*} 2\,{\frac{\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) }{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \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}\,{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}, 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 \sqrt{4 \cos{\left (c + d x \right )} + 3}\, 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 \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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