Optimal. Leaf size=123 \[ \frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0330262, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {17, 2637, 2635, 8} \[ \frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{b \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \cos (c+d x) \, dx}{\sqrt{b \cos (c+d x)}}+\frac{\left (C \sqrt{\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{b \cos (c+d x)}}+\frac{\left (C \sqrt{\cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0882564, size = 61, normalized size = 0.5 \[ \frac{\sqrt{\cos (c+d x)} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.452, size = 63, normalized size = 0.5 \begin{align*}{\frac{C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\,A \left ( dx+c \right ) +2\,B\sin \left ( dx+c \right ) +C \left ( dx+c \right ) }{2\,d}\sqrt{\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{b\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.28292, size = 86, normalized size = 0.7 \begin{align*} \frac{\frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C}{\sqrt{b}} + \frac{8 \, A \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{\sqrt{b}} + \frac{4 \, B \sin \left (d x + c\right )}{\sqrt{b}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02116, size = 598, normalized size = 4.86 \begin{align*} \left [-\frac{{\left (2 \, A + C\right )} \sqrt{-b} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) - 2 \,{\left (C \cos \left (d x + c\right ) + 2 \, B\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, b d \cos \left (d x + c\right )}, \frac{{\left (2 \, A + C\right )} \sqrt{b} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{b} \cos \left (d x + c\right )^{\frac{3}{2}}}\right ) \cos \left (d x + c\right ) +{\left (C \cos \left (d x + c\right ) + 2 \, B\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b d \cos \left (d x + c\right )}\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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt{\cos \left (d x + c\right )}}{\sqrt{b \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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