Optimal. Leaf size=43 \[ -\frac {1}{4} \log \left (\cos \left (\frac {x}{2}\right )-3 \sin \left (\frac {x}{2}\right )\right )+\frac {1}{4} \log \left (3 \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2739, 630, 31}
\begin {gather*} \frac {1}{4} \log \left (3 \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\frac {1}{4} \log \left (\cos \left (\frac {x}{2}\right )-3 \sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 2739
Rubi steps
\begin {align*} \int \frac {1}{3-5 \sin (x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{3-10 x+3 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {3}{4} \text {Subst}\left (\int \frac {1}{-9+3 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{-1+3 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {1}{4} \log \left (1-3 \tan \left (\frac {x}{2}\right )\right )+\frac {1}{4} \log \left (3-\tan \left (\frac {x}{2}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \log \left (\cos \left (\frac {x}{2}\right )-3 \sin \left (\frac {x}{2}\right )\right )+\frac {1}{4} \log \left (3 \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.97, size = 21, normalized size = 0.49 \begin {gather*} -\frac {\text {Log}\left [-1+3 \text {Tan}\left [\frac {x}{2}\right ]\right ]}{4}+\frac {\text {Log}\left [-3+\text {Tan}\left [\frac {x}{2}\right ]\right ]}{4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 22, normalized size = 0.51
method | result | size |
default | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )-3\right )}{4}-\frac {\ln \left (3 \tan \left (\frac {x}{2}\right )-1\right )}{4}\) | \(22\) |
norman | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )-3\right )}{4}-\frac {\ln \left (3 \tan \left (\frac {x}{2}\right )-1\right )}{4}\) | \(22\) |
risch | \(-\frac {\ln \left (-\frac {4}{5}-\frac {3 i}{5}+{\mathrm e}^{i x}\right )}{4}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {4}{5}-\frac {3 i}{5}\right )}{4}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 30, normalized size = 0.70 \begin {gather*} -\frac {1}{4} \, \log \left (\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac {1}{4} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 27, normalized size = 0.63 \begin {gather*} \frac {1}{8} \, \log \left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) - \frac {1}{8} \, \log \left (-4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 20, normalized size = 0.47 \begin {gather*} \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 3 \right )}}{4} - \frac {\log {\left (3 \tan {\left (\frac {x}{2} \right )} - 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 30, normalized size = 0.70 \begin {gather*} 2 \left (\frac {\ln \left |\tan \left (\frac {x}{2}\right )-3\right |}{8}-\frac {\ln \left |3 \tan \left (\frac {x}{2}\right )-1\right |}{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 11, normalized size = 0.26 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{4}-\frac {5}{4}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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