Optimal. Leaf size=340 \[ \frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \tan ^{-1}\left (\frac {\sqrt {\frac {13}{2}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{\sqrt {26} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {2 (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \Pi \left (97-56 \sqrt {3};\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}} \]
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Rubi [A] time = 0.58, antiderivative size = 342, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2136, 219, 2142, 2113, 537, 571, 93, 204} \[ \frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \tan ^{-1}\left (\frac {\sqrt {\frac {13}{2}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{\sqrt {26} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {2 (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \Pi \left (97-56 \sqrt {3};-\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 204
Rule 219
Rule 537
Rule 571
Rule 2113
Rule 2136
Rule 2142
Rubi steps
\begin {align*} \int \frac {1}{(3+x) \sqrt {-1-x^3}} \, dx &=-\frac {\int \frac {1}{\sqrt {-1-x^3}} \, dx}{-2+\sqrt {3}}+\frac {\int \frac {1+\sqrt {3}+x}{(3+x) \sqrt {-1-x^3}} \, dx}{-2+\sqrt {3}}\\ &=\frac {2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2-\sqrt {3}+\left (-2+\sqrt {3}\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\left (-2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ &=\frac {2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-2-\sqrt {3}\right )^2-\left (-2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {\left (4 \sqrt [4]{3} \left (-2-\sqrt {3}\right ) \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-2-\sqrt {3}\right )^2-\left (-2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\left (-2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ &=\frac {2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97-56 \sqrt {3};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {\left (2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (-2-\sqrt {3}\right )^2-\left (-2+\sqrt {3}\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ &=\frac {2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97-56 \sqrt {3};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\left (-2-\sqrt {3}\right )^2+\left (-2+\sqrt {3}\right )^2-\left (\left (-2-\sqrt {3}\right )^2+\left (7-4 \sqrt {3}\right ) \left (-2+\sqrt {3}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {-\frac {\left (-2+\sqrt {3}\right ) \left (1-x+x^2\right )}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ &=\frac {(1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tan ^{-1}\left (\frac {\sqrt {\frac {13}{2}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {26} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97-56 \sqrt {3};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 130, normalized size = 0.38 \[ -\frac {4 \sqrt {2} \sqrt {\frac {i (x+1)}{\sqrt {3}+3 i}} \sqrt {x^2-x+1} \Pi \left (\frac {2 \sqrt {3}}{7 i+\sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {-2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )}{\left (\sqrt {3}+7 i\right ) \sqrt {-x^3-1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{3} - 1}}{x^{4} + 3 \, x^{3} + x + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{3} - 1} {\left (x + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 133, normalized size = 0.39 \[ -\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {7}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {7}{2}+\frac {i \sqrt {3}}{2}\right )} \]
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 \[ \int \frac {1}{\sqrt {-x^{3} - 1} {\left (x + 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 179, normalized size = 0.53 \[ \frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {3}{4}-\frac {\sqrt {3}\,1{}\mathrm {i}}{4};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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 \[ \int \frac {1}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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