Optimal. Leaf size=115 \[ \frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.208582, antiderivative size = 115, 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} \[ \frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}+\frac{4 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a}\\ &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{8 \sqrt{a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac{8 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a^2}\\ &=-\frac{2}{5 d e (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}}-\frac{8 \sqrt{a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac{16 (a+a \sin (c+d x))^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.09899, size = 56, normalized size = 0.49 \[ \frac{2 \left (8 \sin ^2(c+d x)+4 \sin (c+d x)-7\right )}{15 d e \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( -16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,\sin \left ( dx+c \right ) +2 \right ) \cos \left ( dx+c \right ) }{15\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.62497, size = 387, normalized size = 3.37 \begin{align*} -\frac{2 \,{\left (7 \, \sqrt{a} \sqrt{e} - \frac{8 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{25 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{25 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{8 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{7 \, \sqrt{a} \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}}{15 \,{\left (a e^{3} + \frac{3 \, a e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26388, size = 211, normalized size = 1.83 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (8 \, \cos \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) - 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \,{\left (a d e^{3} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d e^{3} \cos \left (d x + c\right )^{2}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}} \sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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