Optimal. Leaf size=44 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \left (e-2 x^2 (d-f)\right )}{\sqrt {d} e}\right )}{4 \sqrt {d} e \sqrt {f}} \]
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Rubi [A] time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6, 1107, 618, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \left (e-2 x^2 (d-f)\right )}{\sqrt {d} e}\right )}{4 \sqrt {d} e \sqrt {f}} \]
Antiderivative was successfully verified.
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Rule 6
Rule 206
Rule 618
Rule 1107
Rubi steps
\begin {align*} \int \frac {x}{e^2+4 e f x^2-4 d f x^4+4 f^2 x^4} \, dx &=\int \frac {x}{e^2+4 e f x^2+\left (-4 d f+4 f^2\right ) x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{e^2+4 e f x+\left (-4 d f+4 f^2\right ) x^2} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{16 d e^2 f-x^2} \, dx,x,4 f \left (e-2 (d-f) x^2\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \left (e+2 (-d+f) x^2\right )}{\sqrt {d} e}\right )}{4 \sqrt {d} e \sqrt {f}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 1.05 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \left (-2 d x^2+e+2 f x^2\right )}{\sqrt {d} e}\right )}{4 \sqrt {d} e \sqrt {f}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 168, normalized size = 3.82 \[ \left [\frac {\sqrt {d f} \log \left (-\frac {4 \, {\left (d^{2} f - 2 \, d f^{2} + f^{3}\right )} x^{4} + d e^{2} + e^{2} f - 4 \, {\left (d e f - e f^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (d e - e f\right )} x^{2} - e^{2}\right )} \sqrt {d f}}{4 \, {\left (d f - f^{2}\right )} x^{4} - 4 \, e f x^{2} - e^{2}}\right )}{8 \, d e f}, \frac {\sqrt {-d f} \arctan \left (-\frac {{\left (2 \, {\left (d - f\right )} x^{2} - e\right )} \sqrt {-d f}}{d e}\right )}{4 \, d e f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 10.05, size = 41, normalized size = 0.93 \[ -\frac {\arctan \left (\frac {2 \, d f x^{2} - 2 \, f^{2} x^{2} - f e}{\sqrt {-d f e^{2}}}\right )}{4 \, \sqrt {-d f e^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 0.95 \[ \frac {\arctanh \left (\frac {-4 e f +2 \left (4 d f -4 f^{2}\right ) x^{2}}{4 \sqrt {d f}\, e}\right )}{4 \sqrt {d f}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.87, size = 67, normalized size = 1.52 \[ \frac {\log \left (\frac {2 \, {\left (d f - f^{2}\right )} x^{2} - e f + \sqrt {d f} e}{2 \, {\left (d f - f^{2}\right )} x^{2} - e f - \sqrt {d f} e}\right )}{8 \, \sqrt {d f} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 199, normalized size = 4.52 \[ \frac {\mathrm {atanh}\left (\frac {16\,d^{3/2}\,f^{3/2}\,x^2}{\frac {8\,e\,f^3}{d}-32\,f^3\,x^2-16\,e\,f^2+16\,d\,f^2\,x^2+8\,d\,e\,f+\frac {16\,f^4\,x^2}{d}}-\frac {32\,\sqrt {d}\,f^{5/2}\,x^2}{\frac {8\,e\,f^3}{d}-32\,f^3\,x^2-16\,e\,f^2+16\,d\,f^2\,x^2+8\,d\,e\,f+\frac {16\,f^4\,x^2}{d}}+\frac {16\,f^{7/2}\,x^2}{\sqrt {d}\,\left (\frac {8\,e\,f^3}{d}-32\,f^3\,x^2-16\,e\,f^2+16\,d\,f^2\,x^2+8\,d\,e\,f+\frac {16\,f^4\,x^2}{d}\right )}\right )}{4\,\sqrt {d}\,e\,\sqrt {f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 75, normalized size = 1.70 \[ - \frac {\frac {\sqrt {\frac {1}{d f}} \log {\left (x^{2} + \frac {- d e \sqrt {\frac {1}{d f}} - e}{2 d - 2 f} \right )}}{8} - \frac {\sqrt {\frac {1}{d f}} \log {\left (x^{2} + \frac {d e \sqrt {\frac {1}{d f}} - e}{2 d - 2 f} \right )}}{8}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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