Optimal. Leaf size=102 \[ \frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0621749, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {17, 3021, 2735, 3770} \[ \frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (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 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \int (B+C \cos (c+d x)) \sec (c+d x) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{\left (B \sqrt{\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \tanh ^{-1}(\sin (c+d x)) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0691534, size = 60, normalized size = 0.59 \[ \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right )}{d (b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.384, size = 72, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -2\,B\cos \left ( dx+c \right ){\it Artanh} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +C\cos \left ( dx+c \right ) \left ( dx+c \right ) +A\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{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 = 2.08213, size = 212, normalized size = 2.08 \begin{align*} \frac{\frac{4 \, A \sqrt{b} \sin \left (2 \, d x + 2 \, c\right )}{b^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + b^{3} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b^{3} \cos \left (2 \, d x + 2 \, c\right ) + b^{3}} + \frac{B{\left (\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )\right )}}{b^{\frac{5}{2}}} + \frac{4 \, C \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac{5}{2}}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19824, size = 886, normalized size = 8.69 \begin{align*} \left [-\frac{2 \, B \sqrt{-b} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sin \left (d x + c\right )}{b \sqrt{\cos \left (d x + c\right )}}\right ) \cos \left (d x + c\right )^{2} + C \sqrt{-b} \cos \left (d x + c\right )^{2} \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 \, \sqrt{b \cos \left (d x + c\right )} A \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b^{3} d \cos \left (d x + c\right )^{2}}, \frac{2 \, C \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 )^{2} + B \sqrt{b} \cos \left (d x + c\right )^{2} \log \left (-\frac{b \cos \left (d x + c\right )^{3} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right ) + 2 \, \sqrt{b \cos \left (d x + c\right )} A \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b^{3} d \cos \left (d x + c\right )^{2}}\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 )} \sqrt{\cos \left (d x + c\right )}}{\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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