Optimal. Leaf size=362 \[ -\frac {\left (c-\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \tanh ^{-1}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {c^2+c d+d^2} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {7+4 \sqrt {3}+\frac {\left (1+\sqrt {3}+x\right )^2}{\left (1-\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \Pi \left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\left (c-d-\sqrt {3} d\right ) \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
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Rubi [A]
time = 0.64, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2168, 2138,
551, 585, 95, 214} \begin {gather*} -\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};\text {ArcSin}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \left (c-\sqrt {3} d-d\right )}-\frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (c-\left (1-\sqrt {3}\right ) d\right ) \tanh ^{-1}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {c^2+c d+d^2}}{\sqrt {d} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \sqrt {c-d}}\right )}{\sqrt {d} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \sqrt {c-d} \sqrt {c^2+c d+d^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 214
Rule 551
Rule 585
Rule 2138
Rule 2168
Rubi steps
\begin {align*} \int \frac {1-\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx &=-\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 ) \text {Subst}\left (\int \frac {1}{\left (-c+\left (1+\sqrt {3}\right ) d+\left (-c+\left (1-\sqrt {3}\right ) d\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 )}{\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}} \left (-c+d+\sqrt {3} d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7+4 \sqrt {3}+x^2} \left (\left (-c+\left (1+\sqrt {3}\right ) d\right )^2-\left (-c+\left (1-\sqrt {3}\right ) d\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} \sqrt {2+\sqrt {3}} \left (-c+\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7+4 \sqrt {3}+x^2} \left (\left (-c+\left (1+\sqrt {3}\right ) d\right )^2-\left (-c+\left (1-\sqrt {3}\right ) d\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 {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 (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\left (c-d-\sqrt {3} d\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}} \left (-c+\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7+4 \sqrt {3}+x} \left (\left (-c+\left (1+\sqrt {3}\right ) d\right )^2-\left (-c+\left (1-\sqrt {3}\right ) d\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 {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 (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\left (c-d-\sqrt {3} d\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}} \left (-c+\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-c+\left (1-\sqrt {3}\right ) d\right )^2-\left (-c+\left (1+\sqrt {3}\right ) d\right )^2-\left (\left (7+4 \sqrt {3}\right ) \left (-c+\left (1-\sqrt {3}\right ) d\right )^2+\left (-c+\left (1+\sqrt {3}\right ) d\right )^2\right ) x^2} \, dx,x,\frac {2 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}}}{\sqrt {7+4 \sqrt {3}+\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 {\left (c-\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \tanh ^{-1}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {c^2+c d+d^2} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {7+4 \sqrt {3}+\frac {\left (1+\sqrt {3}+x\right )^2}{\left (1-\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \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 (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\left (c-d-\sqrt {3} d\right ) \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.41, size = 233, normalized size = 0.64 \begin {gather*} \frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c-\left (-3+\sqrt {3}\right ) d\right ) \sqrt {1-x+x^2} \Pi \left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{3 d \sqrt {-1-x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.25, size = 266, normalized size = 0.73
method | result | size |
default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}+\frac {2 i \left (d \sqrt {3}+c -d \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(266\) |
elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}+\frac {2 i \left (d \sqrt {3}+c -d \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(266\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {x - \sqrt {3} + 1}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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