Optimal. Leaf size=18 \[ -\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (3 \cos (x)-4 \sin (x))\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3153, 212}
\begin {gather*} -\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (3 \cos (x)-4 \sin (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3153
Rubi steps
\begin {align*} \int \frac {1}{4 \cos (x)+3 \sin (x)} \, dx &=-\text {Subst}\left (\int \frac {1}{25-x^2} \, dx,x,3 \cos (x)-4 \sin (x)\right )\\ &=-\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (3 \cos (x)-4 \sin (x))\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(18)=36\).
time = 0.01, size = 43, normalized size = 2.39 \begin {gather*} -\frac {1}{5} \log \left (2 \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {1}{5} \log \left (\cos \left (\frac {x}{2}\right )+2 \sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.04, size = 21, normalized size = 1.17 \begin {gather*} -\frac {\text {Log}\left [-2+\text {Tan}\left [\frac {x}{2}\right ]\right ]}{5}+\frac {\text {Log}\left [1+2 \text {Tan}\left [\frac {x}{2}\right ]\right ]}{5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 22, normalized size = 1.22
method | result | size |
default | \(\frac {\ln \left (2 \tan \left (\frac {x}{2}\right )+1\right )}{5}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-2\right )}{5}\) | \(22\) |
norman | \(\frac {\ln \left (2 \tan \left (\frac {x}{2}\right )+1\right )}{5}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-2\right )}{5}\) | \(22\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}-\frac {3}{5}+\frac {4 i}{5}\right )}{5}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {3}{5}-\frac {4 i}{5}\right )}{5}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (12) = 24\).
time = 0.26, size = 30, normalized size = 1.67 \begin {gather*} \frac {1}{5} \, \log \left (\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \frac {1}{5} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (12) = 24\).
time = 0.34, size = 27, normalized size = 1.50 \begin {gather*} -\frac {1}{10} \, \log \left (\frac {3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac {5}{2}\right ) + \frac {1}{10} \, \log \left (-\frac {3}{2} \, \cos \left (x\right ) + 2 \, \sin \left (x\right ) + \frac {5}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 20, normalized size = 1.11 \begin {gather*} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 2 \right )}}{5} + \frac {\log {\left (2 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{5} \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 = 1.67 \begin {gather*} 2 \left (-\frac {\ln \left |\tan \left (\frac {x}{2}\right )-2\right |}{10}+\frac {\ln \left |2 \tan \left (\frac {x}{2}\right )+1\right |}{10}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 11, normalized size = 0.61 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )}{5}-\frac {3}{5}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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