3.524 \(\int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx\)

Optimal. Leaf size=81 \[ \frac {\log \left (2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}-\frac {\log \left (-2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \]

[Out]

-1/4*ln(e+2*f*x^2-2*x*(-d)^(1/2)*f^(1/2))/(-d)^(1/2)/f^(1/2)+1/4*ln(e+2*f*x^2+2*x*(-d)^(1/2)*f^(1/2))/(-d)^(1/
2)/f^(1/2)

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Rubi [A]  time = 0.06, 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, 1164, 628} \[ \frac {\log \left (2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}-\frac {\log \left (-2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \]

Antiderivative was successfully verified.

[In]

Int[(e - 2*f*x^2)/(e^2 + 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]

[Out]

-Log[e - 2*Sqrt[-d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[-d]*Sqrt[f]) + Log[e + 2*Sqrt[-d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqr
t[-d]*Sqrt[f])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

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]  time = 0.12, size = 191, normalized size = 2.36 \[ \frac {-\frac {\left (\sqrt {d} \sqrt {d+2 e}-d-2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {-\sqrt {d} \sqrt {d+2 e}+d+e}}\right )}{\sqrt {-\sqrt {d} \sqrt {d+2 e}+d+e}}-\frac {\left (\sqrt {d} \sqrt {d+2 e}+d+2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {\sqrt {d} \sqrt {d+2 e}+d+e}}\right )}{\sqrt {\sqrt {d} \sqrt {d+2 e}+d+e}}}{2 \sqrt {2} \sqrt {d} \sqrt {f} \sqrt {d+2 e}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e - 2*f*x^2)/(e^2 + 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]

[Out]

(-(((-d - 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e - Sqrt[d]*Sqrt[d + 2*e]]])/Sqrt[d
+ e - Sqrt[d]*Sqrt[d + 2*e]]) - ((d + 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e + Sq
rt[d]*Sqrt[d + 2*e]]])/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[d + 2*e]*Sqrt[f])

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fricas [A]  time = 0.75, size = 141, normalized size = 1.74 \[ \left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, f^{2} x^{4} - 4 \, {\left (d - e\right )} f x^{2} + e^{2} + 4 \, {\left (2 \, f x^{3} + e x\right )} \sqrt {-d f}}{4 \, f^{2} x^{4} + 4 \, {\left (d + e\right )} f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{d}\right ) - \sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{3} + {\left (2 \, d + e\right )} x\right )} \sqrt {d f}}{d e}\right )}{2 \, d f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*f*x^2+e)/(4*f^2*x^4+4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="fricas")

[Out]

[-1/4*sqrt(-d*f)*log((4*f^2*x^4 - 4*(d - e)*f*x^2 + e^2 + 4*(2*f*x^3 + e*x)*sqrt(-d*f))/(4*f^2*x^4 + 4*(d + e)
*f*x^2 + e^2))/(d*f), -1/2*(sqrt(d*f)*arctan(sqrt(d*f)*x/d) - sqrt(d*f)*arctan((2*f*x^3 + (2*d + e)*x)*sqrt(d*
f)/(d*e)))/(d*f)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*f*x^2+e)/(4*f^2*x^4+4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:inde
x.cc index_m operator + Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argument Valueindex.cc
index_m operator + Error: Bad Argument ValueDone

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maple [B]  time = 0.05, size = 394, normalized size = 4.86 \[ -\frac {\sqrt {2}\, d f \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{4 \sqrt {\left (d +2 e \right ) d \,f^{2}}\, \sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}-\frac {\sqrt {2}\, d f \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{4 \sqrt {\left (d +2 e \right ) d \,f^{2}}\, \sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}-\frac {\sqrt {2}\, e f \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{2 \sqrt {\left (d +2 e \right ) d \,f^{2}}\, \sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}-\frac {\sqrt {2}\, e f \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{2 \sqrt {\left (d +2 e \right ) d \,f^{2}}\, \sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}+\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, f x}{\sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{4 \sqrt {-d f -e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, f x}{\sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}}\right )}{4 \sqrt {d f +e f +\sqrt {\left (d +2 e \right ) d \,f^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*f*x^2+e)/(4*f^2*x^4+4*d*f*x^2+4*e*f*x^2+e^2),x)

[Out]

-1/4*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/(d*f+f*e+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*2^(1/2)/(d*f+f*e+(d*f^2*
(d+2*e))^(1/2))^(1/2))*d-1/2*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/(d*f+f*e+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*
2^(1/2)/(d*f+f*e+(d*f^2*(d+2*e))^(1/2))^(1/2))*e-1/4*2^(1/2)/(d*f+f*e+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*
2^(1/2)/(d*f+f*e+(d*f^2*(d+2*e))^(1/2))^(1/2))-1/4*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/(-d*f-f*e+(d*f^2*(d+2*e))^(
1/2))^(1/2)*arctanh(f*x*2^(1/2)/(-d*f-f*e+(d*f^2*(d+2*e))^(1/2))^(1/2))*d-1/2*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/
(-d*f-f*e+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(-d*f-f*e+(d*f^2*(d+2*e))^(1/2))^(1/2))*e+1/4*2^(1/
2)/(-d*f-f*e+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(-d*f-f*e+(d*f^2*(d+2*e))^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {2 \, f x^{2} - e}{4 \, f^{2} x^{4} + 4 \, d f x^{2} + 4 \, e f x^{2} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*f*x^2+e)/(4*f^2*x^4+4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="maxima")

[Out]

-integrate((2*f*x^2 - e)/(4*f^2*x^4 + 4*d*f*x^2 + 4*e*f*x^2 + e^2), x)

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mupad [B]  time = 3.11, size = 50, normalized size = 0.62 \[ \frac {\mathrm {atan}\left (\frac {2\,f^{3/2}\,x^3+2\,d\,\sqrt {f}\,x+e\,\sqrt {f}\,x}{\sqrt {d}\,e}\right )-\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {d}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e - 2*f*x^2)/(e^2 + 4*f^2*x^4 + 4*d*f*x^2 + 4*e*f*x^2),x)

[Out]

(atan((2*f^(3/2)*x^3 + 2*d*f^(1/2)*x + e*f^(1/2)*x)/(d^(1/2)*e)) - atan((f^(1/2)*x)/d^(1/2)))/(2*d^(1/2)*f^(1/
2))

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sympy [A]  time = 0.56, size = 70, normalized size = 0.86 \[ \frac {\sqrt {- \frac {1}{d f}} \log {\left (- d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} - \frac {\sqrt {- \frac {1}{d f}} \log {\left (d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*f*x**2+e)/(4*f**2*x**4+4*d*f*x**2+4*e*f*x**2+e**2),x)

[Out]

sqrt(-1/(d*f))*log(-d*x*sqrt(-1/(d*f)) + e/(2*f) + x**2)/4 - sqrt(-1/(d*f))*log(d*x*sqrt(-1/(d*f)) + e/(2*f) +
x**2)/4

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