Optimal. Leaf size=20 \[ -\frac {2 \sqrt {d \cos (a+b x)}}{b d} \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2645, 30}
\begin {gather*} -\frac {2 \sqrt {d \cos (a+b x)}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2645
Rubi steps
\begin {align*} \int \frac {\sin (a+b x)}{\sqrt {d \cos (a+b x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {2 \sqrt {d \cos (a+b x)}}{b d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {d \cos (a+b x)}}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 19, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {d \cos \left (b x +a \right )}}{b d}\) | \(19\) |
default | \(-\frac {2 \sqrt {d \cos \left (b x +a \right )}}{b d}\) | \(19\) |
risch | \(-\frac {2 \cos \left (b x +a \right )}{\sqrt {d \cos \left (b x +a \right )}\, b}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{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. 36 vs.
\(2 (17) = 34\).
time = 0.70, size = 36, normalized size = 1.80 \begin {gather*} \begin {cases} - \frac {2 \cos {\left (a + b x \right )}}{b \sqrt {d \cos {\left (a + b x \right )}}} & \text {for}\: b \neq 0 \\\frac {x \sin {\left (a \right )}}{\sqrt {d \cos {\left (a \right )}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.62, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2\,\sqrt {d\,\cos \left (a+b\,x\right )}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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