Optimal. Leaf size=229 \[ -\frac{b (5 A+4 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{b (5 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{5 d} \]
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Rubi [A] time = 0.127084, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {17, 3023, 2748, 2633, 2635, 8} \[ -\frac{b (5 A+4 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{b (5 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{5 d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3023
Rule 2748
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+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+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b C \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac{\left (b \sqrt{b \cos (c+d x)}\right ) \int \cos ^3(c+d x) (5 A+4 C+5 B \cos (c+d x)) \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=\frac{b C \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac{\left (b B \sqrt{b \cos (c+d x)}\right ) \int \cos ^4(c+d x) \, dx}{\sqrt{\cos (c+d x)}}+\frac{\left (b (5 A+4 C) \sqrt{b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{5 \sqrt{\cos (c+d x)}}\\ &=\frac{b B \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{b C \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac{\left (3 b B \sqrt{b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt{\cos (c+d x)}}-\frac{\left (b (5 A+4 C) \sqrt{b \cos (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d \sqrt{\cos (c+d x)}}\\ &=\frac{b (5 A+4 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b B \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac{b B \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{b C \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac{b (5 A+4 C) \sqrt{b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{\left (3 b B \sqrt{b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt{\cos (c+d x)}}\\ &=\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b (5 A+4 C) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b B \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac{b B \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{b C \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac{b (5 A+4 C) \sqrt{b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.317279, size = 109, normalized size = 0.48 \[ \frac{(b \cos (c+d x))^{3/2} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.293, size = 134, normalized size = 0.6 \begin{align*}{\frac{24\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) +30\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +40\,A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+32\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+45\,B\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +80\,A\sin \left ( dx+c \right ) +45\,B \left ( dx+c \right ) +64\,\sin \left ( dx+c \right ) C}{120\,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 = 2.3301, size = 228, normalized size = 1. \begin{align*} \frac{40 \,{\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} + 15 \,{\left (12 \,{\left (d x + c\right )} b + b \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} B \sqrt{b} + 2 \,{\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}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07001, size = 853, normalized size = 3.72 \begin{align*} \left [\frac{45 \, 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 (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \,{\left (5 \, A + 4 \, C\right )} b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )}, \frac{45 \, 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 (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \,{\left (5 \, A + 4 \, C\right )} b\right )} \sqrt{b \cos \left (d x + c\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{120 \, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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