Optimal. Leaf size=101 \[ \frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
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Rubi [A] time = 0.0229203, antiderivative size = 101, 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} \[ \frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2734
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \cos (c+d x) (A+B \cos (c+d x)) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{b B \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.039797, size = 58, normalized size = 0.57 \[ \frac{b \sqrt{b \cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.444, 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 ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( \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 = 2.08354, size = 58, normalized size = 0.57 \begin{align*} \frac{4 \, A b^{\frac{3}{2}} \sin \left (d x + c\right ) +{\left (2 \,{\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} B \sqrt{b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64022, size = 583, normalized size = 5.77 \begin{align*} \left [\frac{B \sqrt{-b} 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 b \cos \left (d x + c\right ) + 2 \, A b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )}, \frac{B b^{\frac{3}{2}} \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 b \cos \left (d x + c\right ) + 2 \, A b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{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 (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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