Optimal. Leaf size=98 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {1}{2} \sin ^{-1}(x+2) \]
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Rubi [A] time = 0.21, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {989, 619, 216, 1028, 986, 12, 1026, 1161, 618, 204, 1027, 206} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {1}{2} \sin ^{-1}(x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 206
Rule 216
Rule 618
Rule 619
Rule 986
Rule 989
Rule 1026
Rule 1027
Rule 1028
Rule 1161
Rubi steps
\begin {align*} \int \frac {\sqrt {-3-4 x-x^2}}{3+4 x+2 x^2} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\sqrt {-3-4 x-x^2}} \, dx\right )-\frac {1}{2} \int \frac {3+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,-4-2 x\right )+\frac {1}{2} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {3}{2} \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\frac {1}{4} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {1}{4} \int -\frac {4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-3 \operatorname {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\int \frac {x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-8 \operatorname {Subst}\left (\int \frac {1+3 x^2}{-4-8 x^2-36 x^4} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right )\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{3}+\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right )\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (-1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right )\\ &=-\frac {1}{2} \sin ^{-1}(2+x)-\frac {\tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.39, size = 159, normalized size = 1.62 \begin {gather*} \frac {1}{4} \left (-i \sqrt {1-2 i \sqrt {2}} \tanh ^{-1}\left (\frac {i \sqrt {2} x+2 x+2 i \sqrt {2}+2}{\sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}}\right )+i \sqrt {1+2 i \sqrt {2}} \tanh ^{-1}\left (\frac {\left (2-i \sqrt {2}\right ) x-2 i \sqrt {2}+2}{\sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}}\right )-2 \sin ^{-1}(x+2)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 80, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x+\frac {3}{\sqrt {2}}}{\sqrt {-x^2-4 x-3}}\right )}{\sqrt {2}}+\tan ^{-1}\left (\frac {\sqrt {-x^2-4 x-3}}{x+3}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 161, normalized size = 1.64 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 171, normalized size = 1.74 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {1}{2} \, \arcsin \left (x + 2\right ) - \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 341, normalized size = 3.48 \begin {gather*} -\frac {\arcsin \left (x +2\right )}{2}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \sqrt {2}}{6}\right )}{3 \sqrt {\frac {\frac {x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-4}{\left (\frac {x}{-x -\frac {3}{2}}+1\right )^{2}}}\, \left (\frac {x}{-x -\frac {3}{2}}+1\right )}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \left (-\arctanh \left (\frac {3 x}{\left (-x -\frac {3}{2}\right ) \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \sqrt {2}}{6}\right )\right )}{12 \sqrt {\frac {\frac {x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-4}{\left (\frac {x}{-x -\frac {3}{2}}+1\right )^{2}}}\, \left (\frac {x}{-x -\frac {3}{2}}+1\right )}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \left (\arctanh \left (\frac {3 x}{\left (-x -\frac {3}{2}\right ) \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \sqrt {2}}{6}\right )\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-4}{\left (\frac {x}{-x -\frac {3}{2}}+1\right )^{2}}}\, \left (\frac {x}{-x -\frac {3}{2}}+1\right )} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {-x^{2} - 4 \, x - 3}}{2 \, x^{2} + 4 \, x + 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {-x^2-4\,x-3}}{2\,x^2+4\,x+3} \,d x \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 {\sqrt {- \left (x + 1\right ) \left (x + 3\right )}}{2 x^{2} + 4 x + 3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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