Optimal. Leaf size=47 \[ \frac {4 \sqrt {a \sin (c+d x)+a}}{a^2 d}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 47, 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} \[ \frac {4 \sqrt {a \sin (c+d x)+a}}{a^2 d}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]
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))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{\sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {2 a}{\sqrt {a+x}}-\sqrt {a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 \sqrt {a+a \sin (c+d x)}}{a^2 d}-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 32, normalized size = 0.68 \[ -\frac {2 (\sin (c+d x)-5) \sqrt {a (\sin (c+d x)+1)}}{3 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 28, normalized size = 0.60 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) - 5\right )}}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.68, size = 56, normalized size = 1.19 \[ \frac {2 \, {\left (3 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{a}\right )}}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 29, normalized size = 0.62 \[ -\frac {2 \sqrt {a +a \sin \left (d x +c \right )}\, \left (\sin \left (d x +c \right )-5\right )}{3 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 36, normalized size = 0.77 \[ -\frac {2 \, {\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 6 \, \sqrt {a \sin \left (d x + c\right ) + a} a\right )}}{3 \, a^{3} d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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