Optimal. Leaf size=49 \[ \frac {4 (a+a \sin (c+d x))^{9/2}}{9 a^2 d}-\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^3 d} \]
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Rubi [A]
time = 0.04, 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 = {2746, 45}
\begin {gather*} \frac {4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\text {Subst}\left (\int (a-x) (a+x)^{7/2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (2 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 (a+a \sin (c+d x))^{9/2}}{9 a^2 d}-\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 41, normalized size = 0.84 \begin {gather*} -\frac {2 (1+\sin (c+d x))^2 (a (1+\sin (c+d x)))^{5/2} (-13+9 \sin (c+d x))}{99 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 31, normalized size = 0.63
method | result | size |
default | \(-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (9 \sin \left (d x +c \right )-13\right )}{99 a^{2} d}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 0.78 \begin {gather*} -\frac {2 \, {\left (9 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 22 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a\right )}}{99 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (41) = 82\).
time = 0.36, size = 88, normalized size = 1.80 \begin {gather*} -\frac {2 \, {\left (23 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2} + {\left (9 \, a^{2} \cos \left (d x + c\right )^{4} - 20 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{99 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs.
\(2 (42) = 84\).
time = 48.11, size = 335, normalized size = 6.84 \begin {gather*} \begin {cases} \frac {8 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{5}{\left (c + d x \right )}}{77 d} + \frac {272 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{4}{\left (c + d x \right )}}{693 d} + \frac {2 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {368 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{3}{\left (c + d x \right )}}{693 d} + \frac {6 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {64 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{231 d} + \frac {6 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {8 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{693 d} + \frac {2 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{7 d} - \frac {16 a^{2} \sqrt {a \sin {\left (c + d x \right )} + a}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{\frac {5}{2}} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.75, size = 76, normalized size = 1.55 \begin {gather*} -\frac {64 \, \sqrt {2} {\left (9 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 11 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{99 \, d} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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