Optimal. Leaf size=49 \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 49, 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} \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x) \sqrt {a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a \sqrt {a+x}-(a+x)^{3/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 (a+a \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {2 (a+a \sin (c+d x))^{5/2}}{5 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 0.69 \[ -\frac {2 (3 \sin (c+d x)-7) (a (\sin (c+d x)+1))^{3/2}}{15 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 40, normalized size = 0.82 \[ \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) + 4\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 75, normalized size = 1.53 \[ \frac {2 \, {\left (15 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{a^{2}}\right )}}{15 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 31, normalized size = 0.63 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (3 \sin \left (d x +c \right )-7\right )}{15 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 75, normalized size = 1.53 \[ \frac {2 \, {\left (15 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{a^{2}}\right )}}{15 \, a d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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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