Optimal. Leaf size=107 \[ \frac{\sin ^5(c+d x) \sqrt{\cos (c+d x)}}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0239209, antiderivative size = 107, 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} \[ \frac{\sin ^5(c+d x) \sqrt{\cos (c+d x)}}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{11}{2}}(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos ^5(c+d x) \, dx}{\sqrt{b \cos (c+d x)}}\\ &=-\frac{\sqrt{\cos (c+d x)} \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{b \cos (c+d x)}}\\ &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{b \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.114257, size = 57, normalized size = 0.53 \[ \frac{\sin (c+d x) \left (3 \sin ^4(c+d x)-10 \sin ^2(c+d x)+15\right ) \sqrt{\cos (c+d x)}}{15 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 52, normalized size = 0.5 \begin{align*}{\frac{ \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8 \right ) \sin \left ( dx+c \right ) }{15\,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 = 1.94109, size = 92, normalized size = 0.86 \begin{align*} \frac{3 \, \sin \left (5 \, d x + 5 \, c\right ) + 25 \, \sin \left (\frac{3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )}{240 \, \sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01629, size = 144, normalized size = 1.35 \begin{align*} \frac{{\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, 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{11}{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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