Optimal. Leaf size=358 \[ -\frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (c-\left (1-\sqrt {3}\right ) d\right ) \tanh ^{-1}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {c^2+c d+d^2}}{\sqrt {d} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \sqrt {c-d}}\right )}{\sqrt {d} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1} \sqrt {c-d} \sqrt {c^2+c d+d^2}}-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1} \left (c-\sqrt {3} d-d\right )} \]
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Rubi [A] time = 1.00, antiderivative size = 360, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2143, 2113, 537, 571, 93, 208} \[ \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\left (1-\sqrt {3}\right ) d\right )^2}{\left (c-\left (1+\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1} \left (c-\sqrt {3} d-d\right )}-\frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (c-\left (1-\sqrt {3}\right ) d\right ) \tanh ^{-1}\left (\frac {2 \sqrt {2+\sqrt {3}} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {c^2+c d+d^2}}{\sqrt {d} \sqrt {\frac {\left (x+\sqrt {3}+1\right )^2}{\left (x-\sqrt {3}+1\right )^2}+4 \sqrt {3}+7} \sqrt {c-d}}\right )}{\sqrt {d} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {x^3+1} \sqrt {c-d} \sqrt {c^2+c d+d^2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rule 537
Rule 571
Rule 2113
Rule 2143
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.53, size = 213, normalized size = 0.59 \[ \frac {2 \sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \left (-\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac {i \sqrt {x^2-x+1} \left (c+\left (\sqrt {3}-1\right ) d\right ) \Pi \left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt {x^3+1}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 275, normalized size = 0.77 \[ \frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, d}-\frac {2 \left (c +\sqrt {3}\, d -d \right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {c}{d}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (\frac {c}{d}-1\right ) d^{2}} \]
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 \[ \int \frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1} {\left (d x + c\right )}}\,{d x} \]
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 \[ \text {Hanged} \]
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 \[ \int \frac {x - \sqrt {3} + 1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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