Optimal. Leaf size=81 \[ -\frac {\log \left (e-2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}+\frac {\log \left (e+2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 1178, 642}
\begin {gather*} \frac {\log \left (2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}-\frac {\log \left (-2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 642
Rule 1178
Rubi steps
\begin {align*} \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx &=\int \frac {e-2 f x^2}{e^2+4 (d+e) f x^2+4 f^2 x^4} \, dx\\ &=-\frac {\int \frac {\frac {\sqrt {-d}}{\sqrt {f}}+2 x}{-\frac {e}{2 f}-\frac {\sqrt {-d} x}{\sqrt {f}}-x^2} \, dx}{4 \sqrt {-d} \sqrt {f}}-\frac {\int \frac {\frac {\sqrt {-d}}{\sqrt {f}}-2 x}{-\frac {e}{2 f}+\frac {\sqrt {-d} x}{\sqrt {f}}-x^2} \, dx}{4 \sqrt {-d} \sqrt {f}}\\ &=-\frac {\log \left (e-2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}+\frac {\log \left (e+2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(81)=162\).
time = 0.08, size = 191, normalized size = 2.36 \begin {gather*} \frac {-\frac {\left (-d-2 e+\sqrt {d} \sqrt {d+2 e}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {d+e-\sqrt {d} \sqrt {d+2 e}}}\right )}{\sqrt {d+e-\sqrt {d} \sqrt {d+2 e}}}-\frac {\left (d+2 e+\sqrt {d} \sqrt {d+2 e}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {d+e+\sqrt {d} \sqrt {d+2 e}}}\right )}{\sqrt {d+e+\sqrt {d} \sqrt {d+2 e}}}}{2 \sqrt {2} \sqrt {d} \sqrt {d+2 e} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs.
\(2(61)=122\).
time = 0.06, size = 193, normalized size = 2.38
method | result | size |
risch | \(-\frac {\ln \left (-2 \sqrt {-d f}\, f \,x^{2}-2 d f x -\sqrt {-d f}\, e \right )}{4 \sqrt {-d f}}+\frac {\ln \left (-2 \sqrt {-d f}\, f \,x^{2}+2 d f x -\sqrt {-d f}\, e \right )}{4 \sqrt {-d f}}\) | \(74\) |
default | \(f^{2} \left (-\frac {\left (d f +2 e f -\sqrt {d \,f^{2} \left (d +2 e \right )}\right ) \sqrt {2}\, \arctanh \left (\frac {f x \sqrt {2}}{\sqrt {-d f -e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )}{4 \sqrt {d \,f^{2} \left (d +2 e \right )}\, f^{2} \sqrt {-d f -e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}+\frac {\left (-d f -2 e f -\sqrt {d \,f^{2} \left (d +2 e \right )}\right ) \sqrt {2}\, \arctan \left (\frac {f x \sqrt {2}}{\sqrt {d f +e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )}{4 \sqrt {d \,f^{2} \left (d +2 e \right )}\, f^{2} \sqrt {d f +e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )\) | \(193\) |
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 [A]
time = 0.41, size = 150, normalized size = 1.85 \begin {gather*} \left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, f^{2} x^{4} - 4 \, d f x^{2} + 4 \, f x^{2} e + 4 \, {\left (2 \, f x^{3} + x e\right )} \sqrt {-d f} + e^{2}}{4 \, f^{2} x^{4} + 4 \, d f x^{2} + 4 \, f x^{2} e + e^{2}}\right )}{4 \, d f}, \frac {\sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{3} + 2 \, d x + x e\right )} \sqrt {d f} e^{\left (-1\right )}}{d}\right ) - \sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{d}\right )}{2 \, d f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 70, normalized size = 0.86 \begin {gather*} \frac {\sqrt {- \frac {1}{d f}} \log {\left (- d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} - \frac {\sqrt {- \frac {1}{d f}} \log {\left (d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [B]
time = 3.11, size = 50, normalized size = 0.62 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2\,f^{3/2}\,x^3+2\,d\,\sqrt {f}\,x+e\,\sqrt {f}\,x}{\sqrt {d}\,e}\right )-\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {d}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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