Optimal. Leaf size=49 \[ \frac {4 (a \sin (c+d x)+a)^{7/2}}{7 a^2 d}-\frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 43} \[ \frac {4 (a \sin (c+d x)+a)^{7/2}}{7 a^2 d}-\frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x) (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 (a+a \sin (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 41, normalized size = 0.84 \[ -\frac {2 (\sin (c+d x)+1)^2 (7 \sin (c+d x)-11) (a (\sin (c+d x)+1))^{3/2}}{63 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 66, normalized size = 1.35 \[ -\frac {2 \, {\left (7 \, a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (5 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) - 16 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{63 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 257, normalized size = 5.24 \[ -\frac {1}{2520} \, \sqrt {2} {\left (\frac {45 \, a \cos \left (\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {210 \, a \cos \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {35 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {126 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {126 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} - \frac {1890 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} - \frac {90 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} - \frac {630 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 31, normalized size = 0.63 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (7 \sin \left (d x +c \right )-11\right )}{63 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 38, normalized size = 0.78 \[ -\frac {2 \, {\left (7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 18 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a\right )}}{63 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 127.62, size = 252, normalized size = 5.14 \[ \begin {cases} \frac {8 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{4}{\left (c + d x \right )}}{45 d} + \frac {152 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{3}{\left (c + d x \right )}}{315 d} + \frac {2 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{21 d} + \frac {4 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 a \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{315 d} + \frac {2 a \sqrt {a \sin {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{5 d} - \frac {16 a \sqrt {a \sin {\left (c + d x \right )} + a}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{\frac {3}{2}} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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