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 = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 43
Rule 2667
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.83, size = 41, normalized size = 0.91 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) + 3\right )}}{a^{3} d \sin \left (d x + c\right ) + a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.45, size = 39, normalized size = 0.87 \[ -\frac {2 \, {\left (\frac {\sqrt {a \sin \left (d x + c\right ) + a}}{a^{3}} + \frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 29, normalized size = 0.64 \[ -\frac {2 \left (3+\sin \left (d x +c \right )\right )}{a^{2} \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 42, normalized size = 0.93 \[ -\frac {2 \, {\left (\frac {\sqrt {a \sin \left (d x + c\right ) + a}}{a^{2}} + \frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a}\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.90, size = 267, normalized size = 5.93 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = \frac {3 \pi }{2} \vee c = - d x + \frac {3 \pi }{2}\right ) \wedge \left (c = - d x + \frac {3 \pi }{2} \vee d = 0\right ) \\\frac {x \cos ^{3}{\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {8 \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {24 \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {16 \sqrt {a \sin {\left (c + d x \right )} + a}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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