Optimal. Leaf size=107 \[ \frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0240382, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {17, 2734} \[ \frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \cos (c+d x) (A+B \cos (c+d x)) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0608085, size = 60, normalized size = 0.56 \[ \frac{\sqrt{\cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 b^2 d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.22, size = 55, normalized size = 0.5 \begin{align*}{\frac{B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +2\,A\sin \left ( dx+c \right ) +B \left ( dx+c \right ) }{2\,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 = 1.97879, size = 54, normalized size = 0.5 \begin{align*} \frac{\frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B}{b^{\frac{5}{2}}} + \frac{4 \, A \sin \left (d x + c\right )}{b^{\frac{5}{2}}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97664, size = 582, normalized size = 5.44 \begin{align*} \left [-\frac{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 (B \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, b^{3} d \cos \left (d x + c\right )}, \frac{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 (B \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, 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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{7}{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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