Optimal. Leaf size=12 \[ -\frac {\cos (x)}{a+a \sin (x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2727}
\begin {gather*} -\frac {\cos (x)}{a \sin (x)+a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rubi steps
\begin {align*} \int \frac {1}{a+a \sin (x)} \, dx &=-\frac {\cos (x)}{a+a \sin (x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(12)=24\).
time = 0.02, size = 29, normalized size = 2.42 \begin {gather*} \frac {2 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{a+a \sin (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 14, normalized size = 1.17
method | result | size |
default | \(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(14\) |
norman | \(-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(14\) |
risch | \(-\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 16, normalized size = 1.33 \begin {gather*} -\frac {2}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 22, normalized size = 1.83 \begin {gather*} -\frac {\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 10, normalized size = 0.83 \begin {gather*} - \frac {2}{a \tan {\left (\frac {x}{2} \right )} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 13, normalized size = 1.08 \begin {gather*} -\frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 13, normalized size = 1.08 \begin {gather*} -\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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