Optimal. Leaf size=73 \[ \frac {2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
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Rubi [A] time = 0.08, 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} \[ \frac {2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{7/2}-4 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^3 d}-\frac {8 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}+\frac {2 (a+a \sin (c+d x))^{13/2}}{13 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 51, normalized size = 0.70 \[ \frac {2 (\sin (c+d x)+1)^3 \left (99 \sin ^2(c+d x)-270 \sin (c+d x)+203\right ) (a (\sin (c+d x)+1))^{3/2}}{1287 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 88, normalized size = 1.21 \[ -\frac {2 \, {\left (99 \, a \cos \left (d x + c\right )^{6} - 14 \, a \cos \left (d x + c\right )^{4} - 32 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (63 \, a \cos \left (d x + c\right )^{4} + 80 \, a \cos \left (d x + c\right )^{2} + 128 \, a\right )} \sin \left (d x + c\right ) - 256 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1287 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 381, normalized size = 5.22 \[ -\frac {1}{288288} \, \sqrt {2} {\left (\frac {819 \, a \cos \left (\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {5148 \, a \cos \left (\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {15015 \, a \cos \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {693 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {4004 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {9009 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {2002 \, a \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 {18018 \, a \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 {180180 \, a \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 {1638 \, a \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 {12870 \, a \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 {60060 \, a \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 )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 41, normalized size = 0.56 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (99 \left (\cos ^{2}\left (d x +c \right )\right )+270 \sin \left (d x +c \right )-302\right )}{1287 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 55, normalized size = 0.75 \[ \frac {2 \, {\left (99 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 468 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a + 572 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{2}\right )}}{1287 \, a^{5} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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