Optimal. Leaf size=119 \[ -\frac{b (A+2 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b C \sin ^5(c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0575295, antiderivative size = 119, 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, 3013, 373} \[ -\frac{b (A+2 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b C \sin ^5(c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\cos (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (A \left (1+\frac{C}{A}\right )-(A+2 C) x^2+C x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\cos (c+d x)}}\\ &=\frac{b (A+C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{b (A+2 C) \sqrt{b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{b C \sqrt{b \cos (c+d x)} \sin ^5(c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.243569, size = 70, normalized size = 0.59 \[ \frac{\sin (c+d x) (b \cos (c+d x))^{3/2} (4 (5 A+7 C) \cos (2 (c+d x))+100 A+3 C \cos (4 (c+d x))+89 C)}{120 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.303, size = 70, normalized size = 0.6 \begin{align*}{\frac{ \left ( 3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+4\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+10\,A+8\,C \right ) \sin \left ( dx+c \right ) }{15\,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 = 2.11192, size = 158, normalized size = 1.33 \begin{align*} \frac{20 \,{\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} A \sqrt{b} +{\left (3 \, b \sin \left (5 \, d x + 5 \, c\right ) + 25 \, b \sin \left (\frac{3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, b \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )\right )} C \sqrt{b}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52038, size = 182, normalized size = 1.53 \begin{align*} \frac{{\left (3 \, C b \cos \left (d x + c\right )^{4} +{\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 2 \,{\left (5 \, A + 4 \, C\right )} b\right )} \sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \sqrt{\cos \left (d x + c\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(-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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