Optimal. Leaf size=101 \[ \frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.0866019, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2647, 2646} \[ \frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac{2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac{1}{5} (5 A+3 B) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{2 a (5 A+3 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 d}+\frac{1}{15} (4 a (5 A+3 B)) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (5 A+3 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 d}\\ \end{align*}
Mathematica [A] time = 0.179546, size = 65, normalized size = 0.64 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)} (2 (5 A+9 B) \cos (c+d x)+50 A+3 B \cos (2 (c+d x))+39 B)}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.117, size = 85, normalized size = 0.8 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{15\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 6\,B \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( -5\,A-15\,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 = 1.81095, size = 126, normalized size = 1.25 \begin{align*} \frac{10 \,{\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} + 3 \,{\left (\sqrt{2} a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sqrt{2} a \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 20 \, \sqrt{2} a \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 = 1.61242, size = 182, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (3 \, B a \cos \left (d x + c\right )^{2} +{\left (5 \, A + 9 \, B\right )} a \cos \left (d x + c\right ) +{\left (25 \, A + 18 \, B\right )} a\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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