Optimal. Leaf size=233 \[ -\frac {38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt {a \sin (c+d x)+a}}-\frac {862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt {a \sin (c+d x)+a}}-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a \sin (c+d x)+a}}+\frac {2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac {6 a \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{143 d}-\frac {1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac {3448 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{45045 d} \]
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Rubi [A] time = 0.72, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt {a \sin (c+d x)+a}}-\frac {862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt {a \sin (c+d x)+a}}-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a \sin (c+d x)+a}}+\frac {2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac {6 a \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{143 d}-\frac {1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac {3448 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{45045 d} \]
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) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\int \sin ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2}\\ &=\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {2 \int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac {5 a^2}{2}-\frac {3}{2} a^2 \sin (c+d x)\right ) \, dx}{13 a^2}\\ &=\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {4 \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {31 a^3}{4}+\frac {19}{4} a^3 \sin (c+d x)\right ) \, dx}{143 a^2}\\ &=-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(431 a) \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1287}\\ &=-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(862 a) \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{3003}\\ &=-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {1724 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{15015}\\ &=-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(862 a) \int \sqrt {a+a \sin (c+d x)} \, dx}{6435}\\ &=-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a+a \sin (c+d x)}}-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}\\ \end {align*}
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Mathematica [A] time = 3.87, size = 120, normalized size = 0.52 \[ -\frac {a \sqrt {a (\sin (c+d x)+1)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (381174 \sin (c+d x)-77665 \sin (3 (c+d x))+3465 \sin (5 (c+d x))-194160 \cos (2 (c+d x))+22680 \cos (4 (c+d x))+281816)}{360360 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 189, normalized size = 0.81 \[ -\frac {2 \, {\left (3465 \, a \cos \left (d x + c\right )^{7} - 4410 \, a \cos \left (d x + c\right )^{6} - 14140 \, a \cos \left (d x + c\right )^{5} + 7330 \, a \cos \left (d x + c\right )^{4} + 15299 \, a \cos \left (d x + c\right )^{3} - 568 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) - {\left (3465 \, a \cos \left (d x + c\right )^{6} + 7875 \, a \cos \left (d x + c\right )^{5} - 6265 \, a \cos \left (d x + c\right )^{4} - 13595 \, a \cos \left (d x + c\right )^{3} + 1704 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) + 4544 \, a\right )} \sin \left (d x + c\right ) + 4544 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45045 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 412, normalized size = 1.77 \[ \frac {1}{1441440} \, \sqrt {2} {\left (\frac {10010 \, 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 {18018 \, 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 {180180 \, 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 {8190 \, 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 {12870 \, 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 {60060 \, 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 {4095 \, 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 {15015 \, 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} + \frac {3465 \, 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 {13}{2} \, d x + \frac {13}{2} \, c\right )}{d} - \frac {10010 \, 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 {9009 \, 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}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 97, normalized size = 0.42 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{2} \left (3465 \left (\sin ^{5}\left (d x +c \right )\right )+11340 \left (\sin ^{4}\left (d x +c \right )\right )+15085 \left (\sin ^{3}\left (d x +c \right )\right )+12930 \left (\sin ^{2}\left (d x +c \right )\right )+10344 \sin \left (d x +c \right )+6896\right )}{45045 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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