Optimal. Leaf size=19 \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
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Rubi [B] time = 0.629963, 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 4401
Rule 12
Rule 2074
Rule 634
Rule 618
Rule 204
Rule 628
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*}
Mathematica [A] time = 0.100563, size = 19, normalized size = 1. \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 35, normalized size = 1.8 \begin{align*} -\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) +\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\tan \left ({\frac{x}{2}} \right ) +2 \right ) -\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45948, size = 72, normalized size = 3.79 \begin{align*} -\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 ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.839, size = 166, normalized size = 8.74 \begin{align*} -\frac{1}{2} \, \log \left (2 \, \cos \left (x\right )^{2} - \frac{1}{2} \,{\left (3 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac{3}{2} \, \cos \left (x\right ) + \frac{1}{2}\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \sin{\left (x \right )} + \cos{\left (x \right )} + 2}{- 2 \sin ^{2}{\left (x \right )} + \sin{\left (x \right )} \cos{\left (x \right )} - 2 \sin{\left (x \right )} + 4 \cos{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12071, size = 49, normalized size = 2.58 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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