Optimal. Leaf size=156 \[ -\frac {64 a^3 \cos ^3(c+d x)}{385 d (a \sin (c+d x)+a)^{3/2}}-\frac {48 a^2 \cos ^3(c+d x)}{385 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 a d}+\frac {4 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{33 d}-\frac {6 a \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{77 d} \]
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Rubi [A] time = 0.43, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2878, 2856, 2674, 2673} \[ -\frac {48 a^2 \cos ^3(c+d x)}{385 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos ^3(c+d x)}{385 d (a \sin (c+d x)+a)^{3/2}}-\frac {2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 a d}+\frac {4 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{33 d}-\frac {6 a \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{77 d} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2856
Rule 2878
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 a d}+\frac {2 \int \cos ^2(c+d x) \left (\frac {5 a}{2}-3 a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{11 a}\\ &=\frac {4 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{33 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 a d}+\frac {3}{11} \int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {6 a \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{77 d}+\frac {4 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{33 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 a d}+\frac {1}{77} (24 a) \int \cos ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {48 a^2 \cos ^3(c+d x)}{385 d \sqrt {a+a \sin (c+d x)}}-\frac {6 a \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{77 d}+\frac {4 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{33 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 a d}+\frac {1}{385} \left (96 a^2\right ) \int \frac {\cos ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {64 a^3 \cos ^3(c+d x)}{385 d (a+a \sin (c+d x))^{3/2}}-\frac {48 a^2 \cos ^3(c+d x)}{385 d \sqrt {a+a \sin (c+d x)}}-\frac {6 a \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{77 d}+\frac {4 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{33 d}-\frac {2 \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 a d}\\ \end {align*}
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Mathematica [A] time = 1.92, size = 110, normalized size = 0.71 \[ -\frac {a \sqrt {a (\sin (c+d x)+1)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (5076 \sin (c+d x)-700 \sin (3 (c+d x))-2280 \cos (2 (c+d x))+105 \cos (4 (c+d x))+4159)}{4620 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 166, normalized size = 1.06 \[ \frac {2 \, {\left (105 \, a \cos \left (d x + c\right )^{6} + 245 \, a \cos \left (d x + c\right )^{5} - 185 \, a \cos \left (d x + c\right )^{4} - 397 \, a \cos \left (d x + c\right )^{3} + 24 \, a \cos \left (d x + c\right )^{2} - 96 \, a \cos \left (d x + c\right ) + {\left (105 \, a \cos \left (d x + c\right )^{5} - 140 \, a \cos \left (d x + c\right )^{4} - 325 \, a \cos \left (d x + c\right )^{3} + 72 \, a \cos \left (d x + c\right )^{2} + 96 \, a \cos \left (d x + c\right ) + 192 \, a\right )} \sin \left (d x + c\right ) - 192 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1155 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 288, normalized size = 1.85 \[ \frac {1}{55440} \, \sqrt {2} {\left (\frac {385 \, 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 {693 \, 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 {6930 \, a \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{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 {315 \, 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 {495 \, 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 {2310 \, 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 {990 \, 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 {770 \, 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 {13860 \, 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}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 87, normalized size = 0.56 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{2} \left (105 \left (\sin ^{4}\left (d x +c \right )\right )+350 \left (\sin ^{3}\left (d x +c \right )\right )+465 \left (\sin ^{2}\left (d x +c \right )\right )+372 \sin \left (d x +c \right )+248\right )}{1155 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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