Optimal. Leaf size=20 \[ \frac {24 x}{25}-\frac {7}{25} \log (4 \cos (x)+3 \sin (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3212}
\begin {gather*} \frac {24 x}{25}-\frac {7}{25} \log (3 \sin (x)+4 \cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3212
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {24 x}{25}-\frac {7}{25} \log (4 \cos (x)+3 \sin (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {24 x}{25}-\frac {7}{25} \log (4 \cos (x)+3 \sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 25, normalized size = 1.25
method | result | size |
risch | \(\frac {24 x}{25}+\frac {7 i x}{25}-\frac {7 \ln \left ({\mathrm e}^{2 i x}+\frac {7}{25}+\frac {24 i}{25}\right )}{25}\) | \(21\) |
default | \(\frac {7 \ln \left (1+\tan ^{2}\left (x \right )\right )}{50}+\frac {24 \arctan \left (\tan \left (x \right )\right )}{25}-\frac {7 \ln \left (3 \tan \left (x \right )+4\right )}{25}\) | \(25\) |
parallelrisch | \(\ln \left (\frac {2}{\left (-2048 \left (\cot \left (x \right )-\csc \left (x \right )+2\right )^{7}\right )^{\frac {1}{25}}}\right )+\ln \left (\frac {1}{\left (-2 \cot \left (x \right )+2 \csc \left (x \right )+1\right )^{\frac {7}{25}}}\right )+\ln \left (\left (\frac {1}{1+\cos \left (x \right )}\right )^{\frac {7}{25}}\right )+\frac {24 x}{25}\) | \(44\) |
norman | \(\frac {\frac {24 x}{25}+\frac {24 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{25}}{1+\tan ^{2}\left (\frac {x}{2}\right )}-\frac {7 \ln \left (\tan \left (\frac {x}{2}\right )-2\right )}{25}-\frac {7 \ln \left (2 \tan \left (\frac {x}{2}\right )+1\right )}{25}+\frac {7 \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{25}\) | \(57\) |
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. 58 vs.
\(2 (16) = 32\).
time = 0.44, size = 58, normalized size = 2.90 \begin {gather*} \frac {48}{25} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) - \frac {7}{25} \, \log \left (\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \frac {7}{25} \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 2\right ) + \frac {7}{25} \, \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 16, normalized size = 0.80 \begin {gather*} \frac {24}{25} \, x - \frac {7}{25} \, \log \left (-2 \, \cos \left (x\right ) - \frac {3}{2} \, \sin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.95 \begin {gather*} \frac {24 x}{25} - \frac {7 \log {\left (3 \sin {\left (x \right )} + 4 \cos {\left (x \right )} \right )}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 23, normalized size = 1.15 \begin {gather*} \frac {24}{25} \, x + \frac {7}{50} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {7}{25} \, \log \left ({\left | 3 \, \tan \left (x\right ) + 4 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 32, normalized size = 1.60 \begin {gather*} \frac {24\,x}{25}-\frac {7\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-1\right )}{25}+\frac {7\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.55 \begin {gather*} -\frac {4 \ln \left (\cos \left (x \right )\right )}{3}+\frac {3 \ln \left (\sin \left (x \right )\right )}{4} \end {gather*}
Warning: Unable to verify antiderivative.
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