Optimal. Leaf size=199 \[ \frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b^2 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.11657, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {17, 3023, 2748, 2635, 8, 2633} \[ \frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b^2 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{9}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \int \cos ^2(c+d x) (4 A+3 C+4 B \cos (c+d x)) \, dx}{4 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{b^2 \sqrt{b \cos (c+d x)}}+\frac{\left ((4 A+3 C) \sqrt{\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{(4 A+3 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}+\frac{\left ((4 A+3 C) \sqrt{\cos (c+d x)}\right ) \int 1 \, dx}{8 b^2 \sqrt{b \cos (c+d x)}}-\frac{\left (B \sqrt{\cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{b^2 d \sqrt{b \cos (c+d x)}}\\ &=\frac{(4 A+3 C) x \sqrt{\cos (c+d x)}}{8 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sqrt{\cos (c+d x)} \sin ^3(c+d x)}{3 b^2 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.143248, size = 95, normalized size = 0.48 \[ \frac{\sqrt{\cos (c+d x)} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.377, size = 114, normalized size = 0.6 \begin{align*}{\frac{6\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +8\,B\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+12\,A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +9\,C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +12\,A \left ( dx+c \right ) +16\,B\sin \left ( dx+c \right ) +9\,C \left ( dx+c \right ) }{24\,d} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( b\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 = 2.33658, size = 157, normalized size = 0.79 \begin{align*} \frac{\frac{24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A}{b^{\frac{5}{2}}} + \frac{3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C}{b^{\frac{5}{2}}} + \frac{8 \, B{\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \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 )}}{b^{\frac{5}{2}}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71896, size = 765, normalized size = 3.84 \begin{align*} \left [-\frac{3 \,{\left (4 \, A + 3 \, C\right )} \sqrt{-b} \cos \left (d x + c\right ) \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 ) - 2 \,{\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 16 \, B\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{48 \, b^{3} d \cos \left (d x + c\right )}, \frac{3 \,{\left (4 \, A + 3 \, C\right )} \sqrt{b} \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 ) \cos \left (d x + c\right ) +{\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 16 \, B\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{24 \, b^{3} d \cos \left (d x + c\right )}\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} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{9}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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