Optimal. Leaf size=19 \[ -\log (1+\cos (x)-2 \sin (x))+\log (3+\cos (x)+\sin (x)) \]
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Rubi [A]
time = 1.28, antiderivative size = 31, normalized size of antiderivative = 1.63, number of steps
used = 32, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4486, 2736,
12, 6857, 648, 632, 210, 642, 209} \begin {gather*} \log \left (\tan ^2\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right )-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 210
Rule 632
Rule 642
Rule 648
Rule 2736
Rule 4486
Rule 6857
Rubi steps
\begin {align*} \int \frac {3+7 \cos (x)+2 \sin (x)}{1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)} \, dx &=\int \left (-\frac {2}{5+\cos (x)}+\frac {17+46 \cos (x)+13 \cos ^2(x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{5+\cos (x)} \, dx\right )+\int \frac {17+46 \cos (x)+13 \cos ^2(x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )} \, dx\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )+\int \left (\frac {17}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )}+\frac {46 \cos (x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )}+\frac {13 \cos ^2(x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )}\right ) \, dx\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )+13 \int \frac {\cos ^2(x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )} \, dx+17 \int \frac {1}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )} \, dx+46 \int \frac {\cos (x)}{(5+\cos (x)) \left (1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)\right )} \, dx\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )+26 \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{8 \left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+34 \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{8 \left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+92 \text {Subst}\left (\int \frac {1-x^4}{8 \left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )+\frac {13}{4} \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {17}{4} \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {23}{2} \text {Subst}\left (\int \frac {1-x^4}{\left (3+2 x^2\right ) \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )+\frac {13}{4} \text {Subst}\left (\int \left (-\frac {9}{154 (-1+2 x)}+\frac {-75-17 x}{77 \left (2+x+x^2\right )}+\frac {25}{14 \left (3+2 x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {17}{4} \text {Subst}\left (\int \left (-\frac {25}{154 (-1+2 x)}+\frac {-3-13 x}{77 \left (2+x+x^2\right )}+\frac {1}{14 \left (3+2 x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {23}{2} \text {Subst}\left (\int \left (-\frac {15}{154 (-1+2 x)}+\frac {29+23 x}{77 \left (2+x+x^2\right )}-\frac {5}{14 \left (3+2 x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {x}{\sqrt {6}}+\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sin (x)}{5+2 \sqrt {6}+\cos (x)}\right )-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )+\frac {13}{308} \text {Subst}\left (\int \frac {-75-17 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {17}{308} \text {Subst}\left (\int \frac {-3-13 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {23}{154} \text {Subst}\left (\int \frac {29+23 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {17}{56} \text {Subst}\left (\int \frac {1}{3+2 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {115}{28} \text {Subst}\left (\int \frac {1}{3+2 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {325}{56} \text {Subst}\left (\int \frac {1}{3+2 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )+\frac {17}{88} \text {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-2 \left (\frac {221}{616} \text {Subst}\left (\int \frac {1+2 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\right )+\frac {529}{308} \text {Subst}\left (\int \frac {1+2 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {115}{44} \text {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {247}{88} \text {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )+\log \left (2+\tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )-\frac {17}{44} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 \tan \left (\frac {x}{2}\right )\right )-\frac {115}{22} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 \tan \left (\frac {x}{2}\right )\right )+\frac {247}{44} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 \tan \left (\frac {x}{2}\right )\right )\\ &=-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )+\log \left (2+\tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 19, normalized size = 1.00 \begin {gather*} -\log (1+\cos (x)-2 \sin (x))+\log (3+\cos (x)+\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 26, normalized size = 1.37
method | result | size |
default | \(\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right )-\ln \left (2 \tan \left (\frac {x}{2}\right )-1\right )\) | \(26\) |
norman | \(\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right )-\ln \left (2 \tan \left (\frac {x}{2}\right )-1\right )\) | \(26\) |
risch | \(\ln \left ({\mathrm e}^{2 i x}+\left (3+3 i\right ) {\mathrm e}^{i x}+i\right )-\ln \left ({\mathrm e}^{2 i x}+\left (\frac {2}{5}-\frac {4 i}{5}\right ) {\mathrm e}^{i x}-\frac {3}{5}-\frac {4 i}{5}\right )\) | \(41\) |
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. 39 vs.
\(2 (19) = 38\).
time = 0.36, size = 39, normalized size = 2.05 \begin {gather*} -\log \left (\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 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. 41 vs.
\(2 (19) = 38\).
time = 0.34, size = 41, normalized size = 2.16 \begin {gather*} -\frac {1}{2} \, \log \left (-\frac {3}{4} \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \frac {1}{2} \, \cos \left (x\right ) + \frac {5}{4}\right ) + \frac {1}{2} \, \log \left (\frac {1}{2} \, {\left (\cos \left (x\right ) + 3\right )} \sin \left (x\right ) + \frac {3}{2} \, \cos \left (x\right ) + \frac {5}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \sin {\left (x \right )} + 7 \cos {\left (x \right )} + 3}{- \sin {\left (x \right )} \cos {\left (x \right )} - 5 \sin {\left (x \right )} + 3 \cos ^{2}{\left (x \right )} + 4 \cos {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 36, normalized size = 1.89 \begin {gather*} 2 \left (-\frac {\ln \left |2 \tan \left (\frac {x}{2}\right )-1\right |}{2}+\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right )}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 23, normalized size = 1.21 \begin {gather*} \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+2\right )-\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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