Optimal. Leaf size=132 \[ \frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d} \]
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Rubi [A] time = 0.174563, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3024, 2751, 2647, 2646} \[ \frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{2 \int (a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (7 A+5 C)-a C \cos (c+d x)\right ) \, dx}{7 a}\\ &=-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{35} (35 A+19 C) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{2 a (35 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}+\frac{1}{105} (4 a (35 A+19 C)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (35 A+19 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 d}-\frac{4 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.243845, size = 75, normalized size = 0.57 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} ((140 A+253 C) \cos (c+d x)+700 A+78 C \cos (2 (c+d x))+15 C \cos (3 (c+d x))+494 C)}{210 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.90594, size = 146, normalized size = 1.11 \begin{align*} \frac{140 \,{\left (\sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 9 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} A \sqrt{a} +{\left (15 \, \sqrt{2} a \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 63 \, \sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 175 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 735 \, \sqrt{2} a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} C \sqrt{a}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3654, size = 223, normalized size = 1.69 \begin{align*} \frac{2 \,{\left (15 \, C a \cos \left (d x + c\right )^{3} + 39 \, C a \cos \left (d x + c\right )^{2} +{\left (35 \, A + 52 \, C\right )} a \cos \left (d x + c\right ) +{\left (175 \, A + 104 \, C\right )} a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\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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