Optimal. Leaf size=101 \[ \frac{2 (5 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d} \]
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Rubi [A] time = 0.201873, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2968, 3023, 2751, 2646} \[ \frac{2 (5 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\int \sqrt{a+a \cos (c+d x)} \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{2 \int \sqrt{a+a \cos (c+d x)} \left (\frac{3 a B}{2}+\frac{1}{2} a (5 A-2 B) \cos (c+d x)\right ) \, dx}{5 a}\\ &=\frac{2 (5 A-2 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{1}{15} (5 A+7 B) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (5 A-2 B) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.185763, size = 64, normalized size = 0.63 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (5 A+4 B) \cos (c+d x)+20 A+3 B \cos (2 (c+d x))+19 B)}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.095, size = 83, normalized size = 0.8 \begin{align*}{\frac{2\,a\sqrt{2}}{15\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 12\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( -10\,A-20\,B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+15\,A+15\,B \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82406, size = 119, normalized size = 1.18 \begin{align*} \frac{10 \,{\left (\sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\left (3 \, \sqrt{2} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 30 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} B \sqrt{a}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34265, size = 170, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (3 \, B \cos \left (d x + c\right )^{2} +{\left (5 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 10 \, A + 8 \, B\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \,{\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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