Optimal. Leaf size=155 \[ \frac{(3 A+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 b d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0600347, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.07, Rules used = {17, 3023, 2734} \[ \frac{(3 A+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 b d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \int \cos (c+d x) (3 A+2 C+3 B \cos (c+d x)) \, dx}{3 b \sqrt{b \cos (c+d x)}}\\ &=\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{(3 A+2 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.169384, size = 75, normalized size = 0.48 \[ \frac{\cos ^{\frac{3}{2}}(c+d x) (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.313, size = 83, normalized size = 0.5 \begin{align*}{\frac{2\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +6\,A\sin \left ( dx+c \right ) +3\,B \left ( dx+c \right ) +4\,\sin \left ( dx+c \right ) C}{6\,d} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( b\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 = 2.33386, size = 108, normalized size = 0.7 \begin{align*} \frac{\frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B}{b^{\frac{3}{2}}} + \frac{C{\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{3}{2}}} + \frac{12 \, A \sin \left (d x + c\right )}{b^{\frac{3}{2}}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05172, size = 667, normalized size = 4.3 \begin{align*} \left [-\frac{3 \, B \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 (2 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 6 \, A + 4 \, C\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{12 \, b^{2} d \cos \left (d x + c\right )}, \frac{3 \, B \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 (2 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 6 \, A + 4 \, C\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6 \, b^{2} 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{5}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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