Optimal. Leaf size=99 \[ \frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0283984, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {17, 2635, 8} \[ \frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \left (A+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{\left (C \sqrt{\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{b \sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt{b \cos (c+d x)}}+\frac{\left (C \sqrt{\cos (c+d x)}\right ) \int 1 \, dx}{2 b \sqrt{b \cos (c+d x)}}\\ &=\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{C \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt{b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0768901, size = 52, normalized size = 0.53 \[ \frac{\cos ^{\frac{3}{2}}(c+d x) (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.273, size = 54, normalized size = 0.6 \begin{align*}{\frac{C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\,A \left ( dx+c \right ) +C \left ( dx+c \right ) }{2\,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 = 2.56736, size = 70, normalized size = 0.71 \begin{align*} \frac{\frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C}{b^{\frac{3}{2}}} + \frac{8 \, A \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac{3}{2}}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66135, size = 475, normalized size = 4.8 \begin{align*} \left [\frac{2 \, \sqrt{b \cos \left (d x + c\right )} C \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) -{\left (2 \, A + C\right )} \sqrt{-b} \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 )}{4 \, b^{2} d}, \frac{\sqrt{b \cos \left (d x + c\right )} C \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) +{\left (2 \, A + 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 )}{2 \, b^{2} d}\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 )} \cos \left (d x + c\right )^{\frac{3}{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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