Optimal. Leaf size=233 \[ \frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d} \]
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Rubi [A] time = 0.64485, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3045, 2976, 2981, 2770, 2774, 216} \[ \frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 3045
Rule 2976
Rule 2981
Rule 2770
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (8 A+3 C)+\frac{1}{2} a (8 B+3 C) \cos (c+d x)\right ) \, dx}{4 a}\\ &=\frac{a (8 B+3 C) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{24 d}+\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \left (\frac{3}{4} a^2 (16 A+8 B+9 C)+\frac{1}{4} a^2 (48 A+56 B+39 C) \cos (c+d x)\right ) \, dx}{12 a}\\ &=\frac{a^2 (48 A+56 B+39 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+3 C) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{24 d}+\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{64} (a (112 A+88 B+75 C)) \int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^2 (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (48 A+56 B+39 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+3 C) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{24 d}+\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{1}{128} (a (112 A+88 B+75 C)) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{a^2 (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (48 A+56 B+39 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+3 C) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{24 d}+\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{(a (112 A+88 B+75 C)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{64 d}\\ &=\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{64 d \sqrt{a+a \cos (c+d x)}}+\frac{a^2 (48 A+56 B+39 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \cos (c+d x)}}+\frac{a (8 B+3 C) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{24 d}+\frac{C \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.950164, size = 145, normalized size = 0.62 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (112 A+88 B+75 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x)} (2 (48 A+88 B+93 C) \cos (c+d x)+336 A+4 (8 B+15 C) \cos (2 (c+d x))+296 B+12 C \cos (3 (c+d x))+285 C)\right )}{384 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 623, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 6.81902, size = 493, normalized size = 2.12 \begin{align*} \frac{{\left (48 \, C a \cos \left (d x + c\right )^{3} + 8 \,{\left (8 \, B + 15 \, C\right )} a \cos \left (d x + c\right )^{2} + 2 \,{\left (48 \, A + 88 \, B + 75 \, C\right )} a \cos \left (d x + c\right ) + 3 \,{\left (112 \, A + 88 \, B + 75 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \,{\left ({\left (112 \, A + 88 \, B + 75 \, C\right )} a \cos \left (d x + c\right ) +{\left (112 \, A + 88 \, B + 75 \, C\right )} a\right )} \sqrt{a} \arctan \left (\frac{\sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right )}{192 \,{\left (d \cos \left (d x + c\right ) + 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(-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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