Optimal. Leaf size=22 \[ -\frac {2}{b d \sqrt {a+b \sin (c+d x)}} \]
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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 = {2747, 32}
\begin {gather*} -\frac {2}{b d \sqrt {a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {2}{b d \sqrt {a+b \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}{b d \sqrt {a+b \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 | \(-\frac {2}{b d \sqrt {a +b \sin \left (d x +c \right )}}\) | \(21\) |
default | \(-\frac {2}{b d \sqrt {a +b \sin \left (d x +c \right )}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.91 \begin {gather*} -\frac {2}{\sqrt {b \sin \left (d x + c\right ) + a} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 32, normalized size = 1.45 \begin {gather*} -\frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b^{2} d \sin \left (d x + c\right ) + a b 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. 56 vs.
\(2 (19) = 38\).
time = 1.00, size = 56, normalized size = 2.55 \begin {gather*} \begin {cases} \frac {x \cos {\left (c \right )}}{a^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{\frac {3}{2}} d} & \text {for}\: b = 0 \\\frac {x \cos {\left (c \right )}}{\left (a + b \sin {\left (c \right )}\right )^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2}{b d \sqrt {a + b \sin {\left (c + d x \right )}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.14, size = 51, normalized size = 2.32 \begin {gather*} -\frac {4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{b\,d\,\left (2\,a^2+4\,a\,b\,\sin \left (c+d\,x\right )+2\,b^2\,{\sin \left (c+d\,x\right )}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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