Optimal. Leaf size=73 \[ \frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 73, 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} \[ \frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}+\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 64, normalized size = 0.88 \[ -\frac {\sqrt {a (\sin (c+d x)+1)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6 (364 \sin (c+d x)+63 \cos (2 (c+d x))-365)}{693 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 68, normalized size = 0.93 \[ \frac {2 \, {\left (7 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.70, size = 189, normalized size = 2.59 \[ \frac {1}{11088} \, \sqrt {2} \sqrt {a} {\left (\frac {77 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {693 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {6930 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {63 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {495 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {2310 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 41, normalized size = 0.56 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (63 \left (\cos ^{2}\left (d x +c \right )\right )+182 \sin \left (d x +c \right )-214\right )}{693 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 55, normalized size = 0.75 \[ \frac {2 \, {\left (63 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 308 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 396 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}\right )}}{693 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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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