Optimal. Leaf size=149 \[ -\frac {13-\frac {27 \sqrt {-x-1}}{\sqrt {x+3}}}{18 \left (-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )}-\frac {2 \left (2-\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )}{9 \left (-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )^2}-\frac {3 \tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 1660, 638, 618, 204} \[ -\frac {13-\frac {27 \sqrt {-x-1}}{\sqrt {x+3}}}{18 \left (-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )}-\frac {2 \left (2-\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )}{9 \left (-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )^2}-\frac {3 \tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 638
Rule 1660
Rubi steps
\begin {align*} \int \frac {1}{\left (x+\sqrt {-3-4 x-x^2}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {2 x \left (1+x^2\right )}{\left (1-2 x+3 x^2\right )^3} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x \left (1+x^2\right )}{\left (1-2 x+3 x^2\right )^3} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=-\frac {2 \left (2-\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )}{9 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\frac {56}{9}+\frac {16 x}{3}}{\left (1-2 x+3 x^2\right )^2} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=-\frac {13-\frac {27 \sqrt {-1-x}}{\sqrt {3+x}}}{18 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )}-\frac {2 \left (2-\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )}{9 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )^2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 x+3 x^2} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=-\frac {13-\frac {27 \sqrt {-1-x}}{\sqrt {3+x}}}{18 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )}-\frac {2 \left (2-\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )}{9 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )^2}-3 \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,-2+\frac {6 \sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=-\frac {13-\frac {27 \sqrt {-1-x}}{\sqrt {3+x}}}{18 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )}-\frac {2 \left (2-\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )}{9 \left (1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}\right )^2}-\frac {3 \tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-1-x}}{\sqrt {3+x}}}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 2.46, size = 914, normalized size = 6.13 \[ \frac {1}{32} \left (\frac {8 (2 x-3)}{\left (2 x^2+4 x+3\right )^2}-\frac {8 \sqrt {-x^2-4 x-3} \left (8 x^3+22 x^2+26 x+15\right )}{\left (2 x^2+4 x+3\right )^2}-12 \sqrt {2} \tan ^{-1}\left (\sqrt {2} (x+1)\right )+\frac {6 \left (2+i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(x+2) \left (2 \left (9+2 i \sqrt {2}\right ) x^2+16 \left (2+i \sqrt {2}\right ) x+3 \left (5+4 i \sqrt {2}\right )\right )}{\left (8 i+6 \sqrt {2}\right ) x^3+\left (-6 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 \sqrt {2}+36 i\right ) x^2+\left (-12 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}-5 \sqrt {2}+40 i\right ) x-9 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}-6 \sqrt {2}+12 i}\right )}{\sqrt {1+2 i \sqrt {2}}}-\frac {6 \left (2 i+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {(x+2) \left (2 \left (9 i+2 \sqrt {2}\right ) x^2+16 \left (2 i+\sqrt {2}\right ) x+3 \left (5 i+4 \sqrt {2}\right )\right )}{\left (-8 i+6 \sqrt {2}\right ) x^3+\left (-6 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 \sqrt {2}-36 i\right ) x^2-12 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3} x-5 \left (8 i+\sqrt {2}\right ) x-3 \left (3 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+2 \sqrt {2}+4 i\right )}\right )}{\sqrt {1-2 i \sqrt {2}}}+\frac {3 \left (2 i+\sqrt {2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt {1-2 i \sqrt {2}}}+\frac {3 \left (-2 i+\sqrt {2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt {1+2 i \sqrt {2}}}-\frac {3 \left (2 i+\sqrt {2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2+2 i \sqrt {2}\right ) x^2+\left (-2 \sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 i \sqrt {2}+4\right ) x-2 \sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+6 i \sqrt {2}+3\right )\right )}{\sqrt {1-2 i \sqrt {2}}}-\frac {3 \left (-2 i+\sqrt {2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2-2 i \sqrt {2}\right ) x^2-2 \left (\sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+4 i \sqrt {2}-2\right ) x-2 \sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}-6 i \sqrt {2}+3\right )\right )}{\sqrt {1+2 i \sqrt {2}}}-\frac {8 (3 x+2)}{2 x^2+4 x+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 171, normalized size = 1.15 \[ -\frac {24 \, x^{3} + 6 \, \sqrt {2} {\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) - 3 \, \sqrt {2} {\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {2} {\left (6 \, x^{2} + 20 \, x + 15\right )} \sqrt {-x^{2} - 4 \, x - 3}}{4 \, {\left (2 \, x^{3} + 11 \, x^{2} + 18 \, x + 9\right )}}\right ) + 64 \, x^{2} + 4 \, {\left (8 \, x^{3} + 22 \, x^{2} + 26 \, x + 15\right )} \sqrt {-x^{2} - 4 \, x - 3} + 60 \, x + 36}{16 \, {\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 367, normalized size = 2.46 \[ -\frac {3}{8} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {6 \, x^{3} + 16 \, x^{2} + 15 \, x + 9}{4 \, {\left (2 \, x^{2} + 4 \, x + 3\right )}^{2}} + \frac {\frac {618 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {1547 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + \frac {2362 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + \frac {2223 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{4}}{{\left (x + 2\right )}^{4}} + \frac {1174 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{5}}{{\left (x + 2\right )}^{5}} + \frac {377 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{6}}{{\left (x + 2\right )}^{6}} + \frac {6 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{7}}{{\left (x + 2\right )}^{7}} + 117}{18 \, {\left (\frac {8 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {14 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + \frac {8 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{4}}{{\left (x + 2\right )}^{4}} + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 14529, normalized size = 97.51 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (x + \sqrt {-x^{2} - 4 \, x - 3}\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (x+\sqrt {-x^2-4\,x-3}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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