Optimal. Leaf size=95 \[ \frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d} \]
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Rubi [A] time = 0.127987, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3024, 2751, 2646} \[ \frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (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{1}{2} a (5 A+3 C)-a C \cos (c+d x)\right ) \, dx}{5 a}\\ &=-\frac{4 C \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{1}{15} (15 A+7 C) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}-\frac{4 C \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.112477, size = 58, normalized size = 0.61 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (30 A+8 C \cos (c+d x)+3 C \cos (2 (c+d x))+19 C)}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 78, 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\,C \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\,C \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+15\,A+7\,C \right ){\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.0009, size = 97, normalized size = 1.02 \begin{align*} \frac{60 \, \sqrt{2} A \sqrt{a} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) +{\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 )} C \sqrt{a}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40498, size = 159, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (3 \, C \cos \left (d x + c\right )^{2} + 4 \, C \cos \left (d x + c\right ) + 15 \, A + 8 \, C\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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