Optimal. Leaf size=19 \[ -\tan ^{-1}\left (\frac {2 \cos (x)-\sin (x)}{\sin (x)+2}\right ) \]
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Rubi [A] time = 3.09, antiderivative size = 38, normalized size of antiderivative = 2.00, number of steps used = 43, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4401, 2648, 12, 6742, 1680, 1673, 1094, 634, 618, 204, 628, 1107} \[ \cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}-\tan ^{-1}\left (\frac {2 \cos (x)-\sin (x)}{\sin (x)+2}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1107
Rule 1673
Rule 1680
Rule 2648
Rule 4401
Rule 6742
Rubi steps
\begin {align*} \int \frac {1+\cos (x)+2 \sin (x)}{3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)} \, dx &=\int \left (\frac {1}{1-\cos (x)}+\frac {2 \left (1+\cos ^2(x)\right )}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )}\right ) \, dx\\ &=2 \int \frac {1+\cos ^2(x)}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )} \, dx+\int \frac {1}{1-\cos (x)} \, dx\\ &=-\frac {\sin (x)}{1-\cos (x)}+2 \int \left (\frac {1}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )}+\frac {\cos ^2(x)}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )}\right ) \, dx\\ &=-\frac {\sin (x)}{1-\cos (x)}+2 \int \frac {1}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )} \, dx+2 \int \frac {\cos ^2(x)}{(-1+\cos (x)) \left (3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)\right )} \, dx\\ &=-\frac {\sin (x)}{1-\cos (x)}+4 \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{8 x^2 \left (-1-x^2-2 x^3-x^4\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+4 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{8 x^2 \left (-1-x^2-2 x^3-x^4\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {\sin (x)}{1-\cos (x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2 \left (-1-x^2-2 x^3-x^4\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2 \left (-1-x^2-2 x^3-x^4\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {\sin (x)}{1-\cos (x)}+\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {-1+2 x}{1+x^2+2 x^3+x^4}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {3+2 x}{1+x^2+2 x^3+x^4}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+2 x}{1+x^2+2 x^3+x^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {3+2 x}{1+x^2+2 x^3+x^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {32 (-1+x)}{17-8 x^2+16 x^4} \, dx,x,\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {32 (1+x)}{17-8 x^2+16 x^4} \, dx,x,\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}+16 \operatorname {Subst}\left (\int \frac {-1+x}{17-8 x^2+16 x^4} \, dx,x,\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )+16 \operatorname {Subst}\left (\int \frac {1+x}{17-8 x^2+16 x^4} \, dx,x,\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}+2 \left (16 \operatorname {Subst}\left (\int \frac {x}{17-8 x^2+16 x^4} \, dx,x,\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}+2 \left (8 \operatorname {Subst}\left (\int \frac {1}{17-8 x+16 x^2} \, dx,x,\left (\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )^2\right )\right )\\ &=\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}-2 \left (16 \operatorname {Subst}\left (\int \frac {1}{-1024-x^2} \, dx,x,-8+32 \left (\frac {1}{2}+\tan \left (\frac {x}{2}\right )\right )^2\right )\right )\\ &=\tan ^{-1}\left (\frac {1}{4} \left (-1+\left (1+2 \tan \left (\frac {x}{2}\right )\right )^2\right )\right )+\cot \left (\frac {x}{2}\right )-\frac {\sin (x)}{1-\cos (x)}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 46, normalized size = 2.42 \[ \frac {1}{2} \tan ^{-1}\left (\frac {\cos (x)+1}{-\sin (x)+\cos (x)-1}\right )-\frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \sec ^2\left (\frac {x}{2}\right ) (-\sin (x)+\cos (x)-1)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 48, normalized size = 2.53 \[ \frac {1}{2} \, \arctan \left (-\frac {3 \, \cos \relax (x)^{2} - 2 \, {\left (3 \, \cos \relax (x) + 1\right )} \sin \relax (x) - 4 \, \cos \relax (x) - 3}{2 \, {\left (2 \, \cos \relax (x)^{2} + {\left (\cos \relax (x) - 3\right )} \sin \relax (x) + 4 \, \cos \relax (x) - 2\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 18, normalized size = 0.95 \[ -\arctan \left (-\tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 13, normalized size = 0.68 \[ \arctan \left (\tan ^{2}\left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x) + 2 \, \sin \relax (x) + 1}{\cos \relax (x)^{2} - 2 \, \cos \relax (x) \sin \relax (x) + 2 \, \sin \relax (x) + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 12, normalized size = 0.63 \[ \mathrm {atan}\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 47.01, size = 94, normalized size = 4.95 \[ \frac {i \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {1}{2} - \frac {\sqrt {1 - 4 i}}{2} \right )}}{2} + \frac {i \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {1}{2} + \frac {\sqrt {1 - 4 i}}{2} \right )}}{2} - \frac {i \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {1}{2} - \frac {\sqrt {1 + 4 i}}{2} \right )}}{2} - \frac {i \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {1}{2} + \frac {\sqrt {1 + 4 i}}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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