Optimal. Leaf size=119 \[ -\frac {256 a^4 \cos (c+d x)}{35 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac {256 a^4 \cos (c+d x)}{35 d \sqrt {a \sin (c+d x)+a}}-\frac {64 a^3 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^{7/2} \, dx &=-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {1}{7} (12 a) \int (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac {24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {1}{35} \left (96 a^2\right ) \int (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {64 a^3 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {1}{35} \left (128 a^3\right ) \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {256 a^4 \cos (c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}-\frac {64 a^3 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}-\frac {24 a^2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 154, normalized size = 1.29 \[ -\frac {a^3 (\sin (c+d x)+1)^3 \sqrt {a (\sin (c+d x)+1)} \left (-1225 \sin \left (\frac {1}{2} (c+d x)\right )+245 \sin \left (\frac {3}{2} (c+d x)\right )+49 \sin \left (\frac {5}{2} (c+d x)\right )-5 \sin \left (\frac {7}{2} (c+d x)\right )+1225 \cos \left (\frac {1}{2} (c+d x)\right )+245 \cos \left (\frac {3}{2} (c+d x)\right )-49 \cos \left (\frac {5}{2} (c+d x)\right )-5 \cos \left (\frac {7}{2} (c+d x)\right )\right )}{140 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 140, normalized size = 1.18 \[ \frac {2 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{4} + 27 \, a^{3} \cos \left (d x + c\right )^{3} - 54 \, a^{3} \cos \left (d x + c\right )^{2} - 204 \, a^{3} \cos \left (d x + c\right ) - 128 \, a^{3} + {\left (5 \, a^{3} \cos \left (d x + c\right )^{3} - 22 \, a^{3} \cos \left (d x + c\right )^{2} - 76 \, a^{3} \cos \left (d x + c\right ) + 128 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 240, normalized size = 2.02 \[ \frac {1}{140} \, \sqrt {2} {\left (\frac {7 \, a^{3} \cos \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {525 \, a^{3} \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {5 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {175 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {70 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} - \frac {42 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {700 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 75, normalized size = 0.63 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right ) \left (5 \left (\sin ^{3}\left (d x +c \right )\right )+27 \left (\sin ^{2}\left (d x +c \right )\right )+71 \sin \left (d x +c \right )+177\right )}{35 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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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