Optimal. Leaf size=65 \[ \frac{b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
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Rubi [A] time = 0.014842, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 2635, 8} \[ \frac{b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt{\cos (c+d x)}}\\ &=\frac{b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0545278, size = 45, normalized size = 0.69 \[ \frac{(2 (c+d x)+\sin (2 (c+d x))) (b \cos (c+d x))^{3/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.25, size = 42, normalized size = 0.7 \begin{align*}{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +dx+c}{2\,d} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78711, size = 38, normalized size = 0.58 \begin{align*} \frac{{\left (2 \,{\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt{b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95001, size = 435, normalized size = 6.69 \begin{align*} \left [\frac{2 \, \sqrt{b \cos \left (d x + c\right )} b \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt{-b} b \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 )}{4 \, d}, \frac{\sqrt{b \cos \left (d x + c\right )} b \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + b^{\frac{3}{2}} \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 )}{2 \, d}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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