Optimal. Leaf size=193 \[ \frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}+\frac {2 a \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {38 a \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt {a \sin (c+d x)+a}}-\frac {76 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{1155 a d}+\frac {152 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3465 d}-\frac {76 a \cos (c+d x)}{495 d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.57, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2879, 2976, 2981, 2770, 2759, 2751, 2646} \[ \frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{11 d}+\frac {2 a \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt {a \sin (c+d x)+a}}-\frac {38 a \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt {a \sin (c+d x)+a}}-\frac {76 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{1155 a d}+\frac {152 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3465 d}-\frac {76 a \cos (c+d x)}{495 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2770
Rule 2879
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\int \sin ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{3/2} \, dx}{a^2}\\ &=\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {2 \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {3 a^2}{2}-\frac {1}{2} a^2 \sin (c+d x)\right ) \, dx}{11 a^2}\\ &=\frac {2 a \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {19}{99} \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {38 a \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}+\frac {38}{231} \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {38 a \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {76 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{1155 a d}+\frac {76 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{1155 a}\\ &=-\frac {38 a \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {152 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3465 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {76 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{1155 a d}+\frac {38}{495} \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {76 a \cos (c+d x)}{495 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a \cos (c+d x) \sin ^3(c+d x)}{693 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sin ^4(c+d x)}{99 d \sqrt {a+a \sin (c+d x)}}+\frac {152 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3465 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{11 d}-\frac {76 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{1155 a d}\\ \end {align*}
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Mathematica [A] time = 1.23, size = 109, normalized size = 0.56 \[ -\frac {\sqrt {a (\sin (c+d x)+1)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (7638 \sin (c+d x)-1330 \sin (3 (c+d x))-3540 \cos (2 (c+d x))+315 \cos (4 (c+d x))+5657)}{13860 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 151, normalized size = 0.78 \[ \frac {2 \, {\left (315 \, \cos \left (d x + c\right )^{6} + 350 \, \cos \left (d x + c\right )^{5} - 500 \, \cos \left (d x + c\right )^{4} - 586 \, \cos \left (d x + c\right )^{3} + 17 \, \cos \left (d x + c\right )^{2} + {\left (315 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4} - 535 \, \cos \left (d x + c\right )^{3} + 51 \, \cos \left (d x + c\right )^{2} + 68 \, \cos \left (d x + c\right ) + 136\right )} \sin \left (d x + c\right ) - 68 \, \cos \left (d x + c\right ) - 136\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3465 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 189, normalized size = 0.98 \[ \frac {1}{55440} \, \sqrt {2} \sqrt {a} {\left (\frac {385 \, \cos \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {693 \, \cos \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {6930 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {315 \, \cos \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {495 \, \cos \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {2310 \, \cos \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.04, size = 85, normalized size = 0.44 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) a \left (\sin \left (d x +c \right )-1\right )^{2} \left (315 \left (\sin ^{4}\left (d x +c \right )\right )+665 \left (\sin ^{3}\left (d x +c \right )\right )+570 \left (\sin ^{2}\left (d x +c \right )\right )+456 \sin \left (d x +c \right )+304\right )}{3465 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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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