Optimal. Leaf size=113 \[ \frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}} \]
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Rubi [A] time = 0.223381, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{4 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{8 \int \frac{(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{11/2}} \, dx}{5 a^2}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{9/2}}-\frac{8 (a+a \sin (c+d x))^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{16 (a+a \sin (c+d x))^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.217391, size = 64, normalized size = 0.57 \[ \frac{2 \left (8 \sin ^2(c+d x)-20 \sin (c+d x)+17\right ) \sec ^5(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{45 d e^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 54, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+40\,\sin \left ( dx+c \right ) -50 \right ) \cos \left ( dx+c \right ) }{45\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59627, size = 381, normalized size = 3.37 \begin{align*} \frac{2 \,{\left (17 \, a^{\frac{5}{2}} \sqrt{e} - \frac{40 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{49 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{49 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{40 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{17 \, a^{\frac{5}{2}} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{45 \,{\left (e^{6} + \frac{3 \, e^{6} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, e^{6} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{e^{6} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49685, size = 254, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (8 \, a^{2} \cos \left (d x + c\right )^{2} + 20 \, a^{2} \sin \left (d x + c\right ) - 25 \, a^{2}\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{45 \,{\left (d e^{6} \cos \left (d x + c\right )^{3} + 2 \, d e^{6} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d e^{6} \cos \left (d x + c\right )\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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