Optimal. Leaf size=74 \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.145952, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx &=-\frac{2 \sqrt{a+a \sin (c+d x)}}{d e (e \cos (c+d x))^{3/2}}+\frac{2 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac{2 \sqrt{a+a \sin (c+d x)}}{d e (e \cos (c+d x))^{3/2}}+\frac{4 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.232978, size = 46, normalized size = 0.62 \[ \frac{2 (2 \sin (c+d x)-1) \sqrt{a (\sin (c+d x)+1)}}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 4\,\sin \left ( dx+c \right ) -2 \right ) \cos \left ( dx+c \right ) }{3\,d}\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) } \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58294, size = 278, normalized size = 3.76 \begin{align*} -\frac{2 \,{\left (\sqrt{a} \sqrt{e} - \frac{4 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{4 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{\sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \,{\left (e^{3} + \frac{2 \, e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{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 = 2.57561, size = 128, normalized size = 1.73 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}{\left (2 \, \sin \left (d x + c\right ) - 1\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \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{\sqrt{a \sin \left (d x + c\right ) + a}}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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