Optimal. Leaf size=80 \[ \frac{b^2 (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b^2 C \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0340368, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {17, 3013} \[ \frac{b^2 (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b^2 C \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3013
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int \cos (c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\cos (c+d x)}}\\ &=\frac{b^2 (A+C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{b^2 C \sqrt{b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.133093, size = 52, normalized size = 0.65 \[ \frac{\sin (c+d x) (b \cos (c+d x))^{5/2} (6 A+C \cos (2 (c+d x))+5 C)}{6 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.261, size = 47, normalized size = 0.6 \begin{align*}{\frac{ \left ( C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,A+2\,C \right ) \sin \left ( dx+c \right ) }{3\,d} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.14171, size = 86, normalized size = 1.08 \begin{align*} \frac{12 \, A b^{\frac{5}{2}} \sin \left (d x + c\right ) +{\left (b^{2} \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b^{2} \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 )} C \sqrt{b}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35093, size = 139, normalized size = 1.74 \begin{align*} \frac{{\left (C b^{2} \cos \left (d x + c\right )^{2} +{\left (3 \, A + 2 \, C\right )} b^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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