Optimal. Leaf size=61 \[ \frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d} \]
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Rubi [A] time = 0.0786756, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3010, 2748, 2639, 2635, 2641} \[ \frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3010
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx &=\int \sqrt{\cos (c+d x)} (B+C \cos (c+d x)) \, dx\\ &=B \int \sqrt{\cos (c+d x)} \, dx+C \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} C \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 C \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.107101, size = 53, normalized size = 0.87 \[ \frac{2 \left (3 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+C \left (F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.297, size = 229, normalized size = 3.8 \begin{align*}{\frac{2}{3\,d}\sqrt{ \left ( 2\, \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 ( -4\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +3\,B\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -C\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) ,\sqrt{2} \right ) +2\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-2\, \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{2\, \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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}\,{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 ({\left (C \cos \left (d x + c\right ) + B\right )} \sqrt{\cos \left (d x + c\right )}, 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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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