Optimal. Leaf size=22 \[ -\frac {2}{a d \sqrt {a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 32}
\begin {gather*} -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {2}{a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} -\frac {2}{a d \sqrt {a+a \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}}\) | \(21\) |
default | \(-\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 0.91 \begin {gather*} -\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 33, normalized size = 1.50 \begin {gather*} -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a^{2} d \sin \left (d x + c\right ) + a^{2} 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. 46 vs.
\(2 (19) = 38\).
time = 0.85, size = 46, normalized size = 2.09 \begin {gather*} \begin {cases} \text {NaN} & \text {for}\: c = \frac {3 \pi }{2} \wedge d = 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{\frac {3}{2}}} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x + \frac {3 \pi }{2} \\- \frac {2}{a d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.72, size = 40, normalized size = 1.82 \begin {gather*} -\frac {\sqrt {2}}{a^{\frac {3}{2}} d \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.88, size = 50, normalized size = 2.27 \begin {gather*} -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d\,\left (2\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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