Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0068127, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2637} \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos (c+d x) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0310402, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.246, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{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 = 1.78478, size = 18, normalized size = 0.56 \begin{align*} \frac{\sin \left (d x + c\right )}{\sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82363, size = 81, normalized size = 2.53 \begin{align*} \frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d \sqrt{\cos \left (d x + c\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 )^{\frac{3}{2}}}{\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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