Optimal. Leaf size=45 \[ -\frac {4}{a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \sqrt {a+a \sin (c+d x)}}{a^3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 45, 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 {2 \sqrt {a \sin (c+d x)+a}}{a^3 d}-\frac {4}{a^2 d \sqrt {a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {a-x}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {2 a}{(a+x)^{3/2}}-\frac {1}{\sqrt {a+x}}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {4}{a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \sqrt {a+a \sin (c+d x)}}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 0.67 \begin {gather*} -\frac {2 (3+\sin (c+d x))}{a^2 d \sqrt {a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 29, normalized size = 0.64
method | result | size |
default | \(-\frac {2 \left (3+\sin \left (d x +c \right )\right )}{a^{2} \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 42, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {a \sin \left (d x + c\right ) + a}}{a^{2}} + \frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a}\right )}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 41, normalized size = 0.91 \begin {gather*} -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) + 3\right )}}{a^{3} d \sin \left (d x + c\right ) + a^{3} 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. 267 vs.
\(2 (41) = 82\).
time = 2.83, size = 267, normalized size = 5.93 \begin {gather*} \begin {cases} \text {NaN} & \text {for}\: \left (c = \frac {3 \pi }{2} \vee c = - d x + \frac {3 \pi }{2}\right ) \wedge \left (c = - d x + \frac {3 \pi }{2} \vee d = 0\right ) \\\frac {x \cos ^{3}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {8 \sqrt {a \sin {\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {24 \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} - \frac {16 \sqrt {a \sin {\left (c + d x \right )} + a}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin {\left (c + d x \right )} + 3 a^{3} d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.80, size = 64, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {\sqrt {2} \sqrt {a}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}}{a^{3} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \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 \frac {{\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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