Optimal. Leaf size=72 \[ \frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0182427, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2633} \[ \frac{b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2633
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\cos (c+d x)}}\\ &=\frac{b \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.11678, size = 45, normalized size = 0.62 \[ \frac{\sin (c+d x) (\cos (2 (c+d x))+5) (b \cos (c+d x))^{3/2}}{6 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.175, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3\,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 = 1.81457, size = 61, normalized size = 0.85 \begin{align*} \frac{{\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} \sqrt{b}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67918, size = 117, normalized size = 1.62 \begin{align*} \frac{{\left (b \cos \left (d x + c\right )^{2} + 2 \, b\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, 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(-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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