Optimal. Leaf size=99 \[ \frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
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Rubi [A] time = 0.0262921, 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} \[ \frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{\left (b^2 C \sqrt{b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{\left (b^2 C \sqrt{b \cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt{\cos (c+d x)}}\\ &=\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 C \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.108027, size = 52, normalized size = 0.53 \[ \frac{(b \cos (c+d x))^{5/2} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, 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 ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( \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 = 1.97368, size = 80, normalized size = 0.81 \begin{align*} \frac{8 \, A b^{\frac{5}{2}} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) +{\left (2 \,{\left (d x + c\right )} b^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} C \sqrt{b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67028, size = 475, normalized size = 4.8 \begin{align*} \left [\frac{2 \, \sqrt{b \cos \left (d x + c\right )} C b^{2} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) +{\left (2 \, A + C\right )} \sqrt{-b} b^{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 )}{4 \, d}, \frac{\sqrt{b \cos \left (d x + c\right )} C b^{2} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) +{\left (2 \, A + C\right )} b^{\frac{5}{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 )}{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 )} \left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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