Optimal. Leaf size=78 \[ \frac{17 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{4 d}+\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d} \]
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Rubi [A] time = 0.101715, antiderivative size = 78, 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, 2661, 2653} \[ \frac{17 F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{4 d}+\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d} \]
Antiderivative was successfully verified.
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Rule 2791
Rule 2752
Rule 2661
Rule 2653
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+d x)|\frac{8}{7}\right )}{4 d}+\frac{17 F\left (\frac{1}{2} (c+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 = 0.0794982, size = 70, normalized size = 0.9 \[ \frac{17 \sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )-21 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+14 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{84 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.297, size = 231, normalized size = 3. \begin{align*} -{\frac{1}{12\,d}\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 32\, \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{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) +3\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \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}+7\, \left ( \sin \left ( 1/2\,dx+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}-1}}}} \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{\cos \left (d x + c\right )^{2}}{\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(-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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