Optimal. Leaf size=38 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A]
time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2118, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2118
Rubi steps
\begin {align*} \int \frac {e-2 f (-1+n) x^n}{e^2+4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx &=-\left (\left (e^2 (1-n)\right ) \text {Subst}\left (\int \frac {1}{e^2+4 d e^2 f (-1+n)^2 x^2} \, dx,x,\frac {x}{e (-1+n)+2 f (-1+n) x^n}\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [F]
time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e-2 f (-1+n) x^n}{e^2+4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs.
\(2(28)=56\).
time = 0.04, size = 78, normalized size = 2.05
method | result | size |
risch | \(-\frac {\ln \left (x^{n}+\frac {2 d f x +\sqrt {-d f}\, e}{2 \sqrt {-d f}\, f}\right )}{4 \sqrt {-d f}}+\frac {\ln \left (x^{n}+\frac {-2 d f x +\sqrt {-d f}\, e}{2 \sqrt {-d f}\, f}\right )}{4 \sqrt {-d f}}\) | \(78\) |
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 = 0.42, size = 146, normalized size = 3.84 \begin {gather*} \left [-\frac {\sqrt {-d f} \log \left (-\frac {4 \, d f x^{2} - 4 \, f^{2} x^{2 \, n} - 4 \, \sqrt {-d f} x e - 4 \, {\left (2 \, \sqrt {-d f} f x + f e\right )} x^{n} - e^{2}}{4 \, d f x^{2} + 4 \, f^{2} x^{2 \, n} + 4 \, f x^{n} e + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {2 \, \sqrt {d f} f x^{n} + \sqrt {d f} e}{2 \, d f x}\right )}{2 \, d f}\right ] \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 {e - 2 f n x^{n} + 2 f x^{n}}{4 d f x^{2} + e^{2} + 4 e f x^{n} + 4 f^{2} x^{2 n}}\, 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 [B]
time = 3.42, size = 196, normalized size = 5.16 \begin {gather*} \frac {\ln \left (-\frac {e+2\,f\,x^n-2\,f\,n\,x^n}{4\,f^2}-\frac {e^2\,n-4\,d\,f\,x^2+4\,d\,f\,n\,x^2+2\,e\,f\,n\,x^n}{8\,\sqrt {-d}\,f^{5/2}\,x}\right )}{4\,\sqrt {-d}\,\sqrt {f}}-\frac {\mathrm {atan}\left (\frac {x\,\left (8\,d\,f\,n^2-16\,d\,f\,n+8\,d\,f\right )}{4\,\sqrt {d}\,\sqrt {f}\,\left (e\,n-e\,n^2\right )}\right )}{2\,\sqrt {d}\,\sqrt {f}}-\frac {\ln \left (\frac {e^2\,n-4\,d\,f\,x^2+4\,d\,f\,n\,x^2+2\,e\,f\,n\,x^n}{8\,\sqrt {-d}\,f^{5/2}\,x}-\frac {e+2\,f\,x^n-2\,f\,n\,x^n}{4\,f^2}\right )}{4\,\sqrt {-d}\,\sqrt {f}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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