Optimal. Leaf size=45 \[ -\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0676261, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2667, 43} \[ -\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{2 a}{(a+x)^{3/2}}-\frac{1}{\sqrt{a+x}}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{4}{a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \sqrt{a+a \sin (c+d x)}}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0544934, size = 30, normalized size = 0.67 \[ -\frac{2 (\sin (c+d x)+3)}{a^2 d \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 29, normalized size = 0.6 \begin{align*} -2\,{\frac{3+\sin \left ( dx+c \right ) }{{a}^{2}\sqrt{a+a\sin \left ( dx+c \right ) }d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942827, size = 57, normalized size = 1.27 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{a \sin \left (d x + c\right ) + a}}{a^{2}} + \frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a}\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13332, size = 104, normalized size = 2.31 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sin \left (d x + c\right ) + 3\right )}}{a^{3} d \sin \left (d x + c\right ) + a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.4713, size = 267, normalized size = 5.93 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: \left (c = \frac{3 \pi }{2} \vee c = - d x + \frac{3 \pi }{2}\right ) \wedge \left (c = - d x + \frac{3 \pi }{2} \vee d = 0\right ) \\\frac{x \cos ^{3}{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{\frac{5}{2}}} & \text{for}\: d = 0 \\- \frac{8 \sqrt{a \sin{\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{24 \sqrt{a \sin{\left (c + d x \right )} + a} \sin{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{2 \sqrt{a \sin{\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{16 \sqrt{a \sin{\left (c + d x \right )} + a}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1691, size = 49, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (\sqrt{a \sin \left (d x + c\right ) + a} + \frac{2 \, a}{\sqrt{a \sin \left (d x + c\right ) + a}}\right )}}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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