Optimal. Leaf size=19 \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
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Rubi [B] time = 0.63, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 25, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4401, 12, 2074, 634, 618, 204, 628} \[ \log \left (\tan ^2\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right )-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )-\log \left (\tan \left (\frac {x}{2}\right )+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 628
Rule 634
Rule 2074
Rule 4401
Rubi steps
\begin {align*} \int \frac {2+\cos (x)+5 \sin (x)}{4 \cos (x)-2 \sin (x)+\cos (x) \sin (x)-2 \sin ^2(x)} \, dx &=\int \left (\frac {\cos (x)}{4 \cos (x)-2 \sin (x)+\cos (x) \sin (x)-2 \sin ^2(x)}-\frac {2}{-4 \cos (x)+2 \sin (x)-\cos (x) \sin (x)+2 \sin ^2(x)}-\frac {5 \sin (x)}{-4 \cos (x)+2 \sin (x)-\cos (x) \sin (x)+2 \sin ^2(x)}\right ) \, dx\\ &=-\left (2 \int \frac {1}{-4 \cos (x)+2 \sin (x)-\cos (x) \sin (x)+2 \sin ^2(x)} \, dx\right )-5 \int \frac {\sin (x)}{-4 \cos (x)+2 \sin (x)-\cos (x) \sin (x)+2 \sin ^2(x)} \, dx+\int \frac {\cos (x)}{4 \cos (x)-2 \sin (x)+\cos (x) \sin (x)-2 \sin ^2(x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1-x}{2 \left (2-3 x-x^2-2 x^3\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-4 \operatorname {Subst}\left (\int \frac {-1-x^2}{2 \left (2-x-4 x^2-3 x^3-2 x^4\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-10 \operatorname {Subst}\left (\int \frac {x}{-2+x+4 x^2+3 x^3+2 x^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {-1-x^2}{2-x-4 x^2-3 x^3-2 x^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\right )-10 \operatorname {Subst}\left (\int \left (\frac {1}{6 (1+x)}+\frac {4}{33 (-1+2 x)}+\frac {-2-5 x}{22 \left (2+x+x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\operatorname {Subst}\left (\int \frac {1-x}{2-3 x-x^2-2 x^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {20}{33} \log \left (1-2 \tan \left (\frac {x}{2}\right )\right )-\frac {5}{3} \log \left (1+\tan \left (\frac {x}{2}\right )\right )-\frac {5}{11} \operatorname {Subst}\left (\int \frac {-2-5 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-2 \operatorname {Subst}\left (\int \left (-\frac {1}{3 (1+x)}+\frac {10}{33 (-1+2 x)}+\frac {3+2 x}{11 \left (2+x+x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\operatorname {Subst}\left (\int \left (-\frac {2}{11 (-1+2 x)}+\frac {7+x}{11 \left (2+x+x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )-\log \left (1+\tan \left (\frac {x}{2}\right )\right )+\frac {1}{11} \operatorname {Subst}\left (\int \frac {7+x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {2}{11} \operatorname {Subst}\left (\int \frac {3+2 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {5}{22} \operatorname {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {25}{22} \operatorname {Subst}\left (\int \frac {1+2 x}{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 (1+\tan \left (\frac {x}{2}\right )\right )+\frac {25}{22} \log \left (2+\tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )+\frac {1}{22} \operatorname {Subst}\left (\int \frac {1+2 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {2}{11} \operatorname {Subst}\left (\int \frac {1+2 x}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {4}{11} \operatorname {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {5}{11} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 \tan \left (\frac {x}{2}\right )\right )+\frac {13}{22} \operatorname {Subst}\left (\int \frac {1}{2+x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {5 x}{22 \sqrt {7}}-\frac {5 \tan ^{-1}\left (\frac {\cos (x)-\sin (x)}{3+\sqrt {7}+\cos (x)+\sin (x)}\right )}{11 \sqrt {7}}-\log \left (1-2 \tan \left (\frac {x}{2}\right )\right )-\log \left (1+\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 {8}{11} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 \tan \left (\frac {x}{2}\right )\right )-\frac {13}{11} \operatorname {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 (1+\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 = 0.10, size = 19, normalized size = 1.00 \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 43, normalized size = 2.26 \[ -\frac {1}{2} \, \log \left (2 \, \cos \relax (x)^{2} - \frac {1}{2} \, {\left (3 \, \cos \relax (x) - 1\right )} \sin \relax (x) - \frac {3}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (\frac {1}{2} \, {\left (\cos \relax (x) + 3\right )} \sin \relax (x) + \frac {3}{2} \, \cos \relax (x) + \frac {5}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 36, normalized size = 1.89 \[ \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) + 2\right ) - \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 35, normalized size = 1.84 \[ -\ln \left (\tan \left (\frac {x}{2}\right )+1\right )-\ln \left (2 \tan \left (\frac {x}{2}\right )-1\right )+\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 53, normalized size = 2.79 \[ -\log \left (\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) + \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + \frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 2\right ) - \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 32, normalized size = 1.68 \[ -2\,\mathrm {atanh}\left (\frac {\frac {252\,\mathrm {tan}\left (\frac {x}{2}\right )}{19}+\frac {1260}{19}}{19\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {x}{2}\right )-32}+\frac {37}{19}\right ) \]
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 \[ \int \frac {5 \sin {\relax (x )} + \cos {\relax (x )} + 2}{- 2 \sin ^{2}{\relax (x )} + \sin {\relax (x )} \cos {\relax (x )} - 2 \sin {\relax (x )} + 4 \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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