Optimal. Leaf size=22 \[ \frac {2 \sqrt {a+b \sin (c+d x)}}{b d} \]
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Rubi [A]
time = 0.02, 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 \sqrt {a+b \sin (c+d x)}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+x}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {2 \sqrt {a+b \sin (c+d x)}}{b d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a+b \sin (c+d x)}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {2 \sqrt {a +b \sin \left (d x +c \right )}}{b d}\) | \(21\) |
default | \(\frac {2 \sqrt {a +b \sin \left (d x +c \right )}}{b d}\) | \(21\) |
risch | \(-\frac {i \sqrt {2}\, \left (2 i a +2 i b \sin \left (d x +c \right )\right )}{\sqrt {2 b \sin \left (d x +c \right )+2 a}\, d b}\) | \(43\) |
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 {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 = 0.34, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (17) = 34\).
time = 0.55, size = 54, normalized size = 2.45 \begin {gather*} \begin {cases} \frac {x \cos {\left (c \right )}}{\sqrt {a}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{\sqrt {a} d} & \text {for}\: b = 0 \\\frac {x \cos {\left (c \right )}}{\sqrt {a + b \sin {\left (c \right )}}} & \text {for}\: d = 0 \\\frac {2 \sqrt {a + b \sin {\left (c + d x \right )}}}{b d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.40, 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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Mupad [B]
time = 6.22, size = 20, normalized size = 0.91 \begin {gather*} \frac {2\,\sqrt {a+b\,\sin \left (c+d\,x\right )}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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