Optimal. Leaf size=156 \[ -\frac {256 a^4 \cos ^5(c+d x)}{5005 d (a \sin (c+d x)+a)^{5/2}}-\frac {64 a^3 \cos ^5(c+d x)}{1001 d (a \sin (c+d x)+a)^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{143 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}-\frac {6 a \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{143 d} \]
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Rubi [A] time = 0.32, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2856, 2674, 2673} \[ -\frac {8 a^2 \cos ^5(c+d x)}{143 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos ^5(c+d x)}{1001 d (a \sin (c+d x)+a)^{3/2}}-\frac {256 a^4 \cos ^5(c+d x)}{5005 d (a \sin (c+d x)+a)^{5/2}}-\frac {2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}-\frac {6 a \cos ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{143 d} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2856
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {2 \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {3}{13} \int \cos ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {6 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {1}{143} (36 a) \int \cos ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {8 a^2 \cos ^5(c+d x)}{143 d \sqrt {a+a \sin (c+d x)}}-\frac {6 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {1}{143} \left (32 a^2\right ) \int \frac {\cos ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {64 a^3 \cos ^5(c+d x)}{1001 d (a+a \sin (c+d x))^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{143 d \sqrt {a+a \sin (c+d x)}}-\frac {6 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {\left (128 a^3\right ) \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{1001}\\ &=-\frac {256 a^4 \cos ^5(c+d x)}{5005 d (a+a \sin (c+d x))^{5/2}}-\frac {64 a^3 \cos ^5(c+d x)}{1001 d (a+a \sin (c+d x))^{3/2}}-\frac {8 a^2 \cos ^5(c+d x)}{143 d \sqrt {a+a \sin (c+d x)}}-\frac {6 a \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {2 \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}\\ \end {align*}
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Mathematica [A] time = 5.18, size = 110, normalized size = 0.71 \[ -\frac {a \sqrt {a (\sin (c+d x)+1)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 (28230 \sin (c+d x)-3290 \sin (3 (c+d x))-12600 \cos (2 (c+d x))+385 \cos (4 (c+d x))+19559)}{20020 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 189, normalized size = 1.21 \[ \frac {2 \, {\left (385 \, a \cos \left (d x + c\right )^{7} - 490 \, a \cos \left (d x + c\right )^{6} - 1015 \, a \cos \left (d x + c\right )^{5} + 20 \, a \cos \left (d x + c\right )^{4} - 32 \, a \cos \left (d x + c\right )^{3} + 64 \, a \cos \left (d x + c\right )^{2} - 256 \, a \cos \left (d x + c\right ) - {\left (385 \, a \cos \left (d x + c\right )^{6} + 875 \, a \cos \left (d x + c\right )^{5} - 140 \, a \cos \left (d x + c\right )^{4} - 160 \, a \cos \left (d x + c\right )^{3} - 192 \, a \cos \left (d x + c\right )^{2} - 256 \, a \cos \left (d x + c\right ) - 512 \, a\right )} \sin \left (d x + c\right ) - 512 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5005 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.81, size = 412, normalized size = 2.64 \[ -\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 {54054 \, 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 {38610 \, 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 = 0.86, size = 87, normalized size = 0.56 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{3} \left (385 \left (\sin ^{4}\left (d x +c \right )\right )+1645 \left (\sin ^{3}\left (d x +c \right )\right )+2765 \left (\sin ^{2}\left (d x +c \right )\right )+2295 \sin \left (d x +c \right )+918\right )}{5005 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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