Integrand size = 37, antiderivative size = 38 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \] Output:
1/2*arctanh(2*d^(1/2)*f^(1/2)*x/(2*f*x^2+e))/d^(1/2)/f^(1/2)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 233, normalized size of antiderivative = 6.13 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {-\frac {\left (-i d+2 i e+\sqrt {d} \sqrt {-d+2 e}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {-d+e-i \sqrt {d} \sqrt {-d+2 e}}}\right )}{\sqrt {-d+e-i \sqrt {d} \sqrt {-d+2 e}}}-\frac {\left (i d-2 i e+\sqrt {d} \sqrt {-d+2 e}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {-d+e+i \sqrt {d} \sqrt {-d+2 e}}}\right )}{\sqrt {-d+e+i \sqrt {d} \sqrt {-d+2 e}}}}{2 \sqrt {2} \sqrt {d} \sqrt {-d+2 e} \sqrt {f}} \] Input:
Integrate[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]
Output:
(-((((-I)*d + (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x) /Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2*e]]])/Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2*e]]) - ((I*d - (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sqrt [f]*x)/Sqrt[-d + e + I*Sqrt[d]*Sqrt[-d + 2*e]]])/Sqrt[-d + e + I*Sqrt[d]*S qrt[-d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[-d + 2*e]*Sqrt[f])
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6, 1478, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e-2 f x^2}{-4 d f x^2+e^2+4 e f x^2+4 f^2 x^4} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e-2 f x^2}{f x^2 (4 e-4 d)+e^2+4 f^2 x^4}dx\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle -\frac {\int -\frac {2 \left (\sqrt {d}-2 \sqrt {f} x\right )}{\sqrt {f} \left (2 x^2-\frac {2 \sqrt {d} x}{\sqrt {f}}+\frac {e}{f}\right )}dx}{4 \sqrt {d} \sqrt {f}}-\frac {\int -\frac {2 \left (2 \sqrt {f} x+\sqrt {d}\right )}{\sqrt {f} \left (2 x^2+\frac {2 \sqrt {d} x}{\sqrt {f}}+\frac {e}{f}\right )}dx}{4 \sqrt {d} \sqrt {f}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {d}-2 \sqrt {f} x}{2 x^2-\frac {2 \sqrt {d} x}{\sqrt {f}}+\frac {e}{f}}dx}{2 \sqrt {d} f}+\frac {\int \frac {2 \sqrt {f} x+\sqrt {d}}{2 x^2+\frac {2 \sqrt {d} x}{\sqrt {f}}+\frac {e}{f}}dx}{2 \sqrt {d} f}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\log \left (2 \sqrt {d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {d} \sqrt {f}}-\frac {\log \left (-2 \sqrt {d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {d} \sqrt {f}}\) |
Input:
Int[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]
Output:
-1/4*Log[e - 2*Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(Sqrt[d]*Sqrt[f]) + Log[e + 2* Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[d]*Sqrt[f])
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(28)=56\).
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74
method | result | size |
risch | \(\frac {\ln \left (2 \sqrt {d f}\, f \,x^{2}+2 d f x +\sqrt {d f}\, e \right )}{4 \sqrt {d f}}-\frac {\ln \left (2 \sqrt {d f}\, f \,x^{2}-2 d f x +\sqrt {d f}\, e \right )}{4 \sqrt {d f}}\) | \(66\) |
default | \(f^{2} \left (-\frac {\left (-d f +2 e f -\sqrt {d \,f^{2} \left (d -2 e \right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {f x \sqrt {2}}{\sqrt {d f -e f +\sqrt {d \,f^{2} \left (d -2 e \right )}}}\right )}{4 f^{2} \sqrt {d \,f^{2} \left (d -2 e \right )}\, \sqrt {d f -e f +\sqrt {d \,f^{2} \left (d -2 e \right )}}}+\frac {\left (d f -2 e f -\sqrt {d \,f^{2} \left (d -2 e \right )}\right ) \sqrt {2}\, \arctan \left (\frac {f x \sqrt {2}}{\sqrt {-d f +e f +\sqrt {d \,f^{2} \left (d -2 e \right )}}}\right )}{4 f^{2} \sqrt {d \,f^{2} \left (d -2 e \right )}\, \sqrt {-d f +e f +\sqrt {d \,f^{2} \left (d -2 e \right )}}}\right )\) | \(193\) |
Input:
int((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x,method=_RETURNVERBO SE)
Output:
1/4/(d*f)^(1/2)*ln(2*(d*f)^(1/2)*f*x^2+2*d*f*x+(d*f)^(1/2)*e)-1/4/(d*f)^(1 /2)*ln(2*(d*f)^(1/2)*f*x^2-2*d*f*x+(d*f)^(1/2)*e)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.84 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\left [\frac {\sqrt {d f} \log \left (\frac {4 \, f^{2} x^{4} + 4 \, {\left (d + e\right )} f x^{2} + e^{2} + 4 \, {\left (2 \, f x^{3} + e x\right )} \sqrt {d f}}{4 \, f^{2} x^{4} - 4 \, {\left (d - e\right )} f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (\frac {\sqrt {-d f} x}{d}\right ) - \sqrt {-d f} \arctan \left (\frac {{\left (2 \, f x^{3} - {\left (2 \, d - e\right )} x\right )} \sqrt {-d f}}{d e}\right )}{2 \, d f}\right ] \] Input:
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="f ricas")
Output:
[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)*ar ctan(sqrt(-d*f)*x/d) - sqrt(-d*f)*arctan((2*f*x^3 - (2*d - e)*x)*sqrt(-d*f )/(d*e)))/(d*f)]
Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=- \frac {\sqrt {\frac {1}{d f}} \log {\left (- d x \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} + \frac {\sqrt {\frac {1}{d f}} \log {\left (d x \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} \] Input:
integrate((-2*f*x**2+e)/(4*f**2*x**4-4*d*f*x**2+4*e*f*x**2+e**2),x)
Output:
-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
\[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\int { -\frac {2 \, f x^{2} - e}{4 \, f^{2} x^{4} - 4 \, d f x^{2} + 4 \, e f x^{2} + e^{2}} \,d x } \] Input:
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="m axima")
Output:
-integrate((2*f*x^2 - e)/(4*f^2*x^4 - 4*d*f*x^2 + 4*e*f*x^2 + e^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 415, normalized size of antiderivative = 10.92 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {{\left (\sqrt {d^{2} - 2 \, d e} e f {\left | f \right |} - {\left (d^{2} - 2 \, d e\right )} f^{2} + {\left (d^{2} - 3 \, d e + 2 \, e^{2}\right )} f^{2}\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {4 \, d f - 4 \, e f + \sqrt {-16 \, e^{2} f^{2} + 16 \, {\left (d f - e f\right )}^{2}}}{f^{2}}}}\right )}{2 \, {\left (\sqrt {d^{2} - 2 \, d e} \sqrt {-d f + 2 \, e f} e f + \sqrt {d^{2} - 2 \, d e} \sqrt {-d f} e f\right )} {\left | f \right |}} - \frac {{\left (\sqrt {-d f + 2 \, e f} d^{2} f^{2} + \sqrt {-d f} d^{2} f^{2} - 3 \, \sqrt {-d f + 2 \, e f} d e f^{2} - 3 \, \sqrt {-d f} d e f^{2} + 2 \, \sqrt {-d f + 2 \, e f} e^{2} f^{2} + 2 \, \sqrt {-d f} e^{2} f^{2} - {\left (\sqrt {-d f + 2 \, e f} d^{2} + \sqrt {-d f} d^{2} - 2 \, \sqrt {-d f + 2 \, e f} d e - 2 \, \sqrt {-d f} d e\right )} f^{2} - {\left (\sqrt {d^{2} - 2 \, d e} \sqrt {-d f + 2 \, e f} e f + \sqrt {d^{2} - 2 \, d e} \sqrt {-d f} e f\right )} {\left | f \right |}\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {4 \, d f - 4 \, e f - \sqrt {-16 \, e^{2} f^{2} + 16 \, {\left (d f - e f\right )}^{2}}}{f^{2}}}}\right )}{4 \, \sqrt {d^{2} - 2 \, d e} e^{2} f^{2} {\left | f \right |}} \] Input:
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="g iac")
Output:
1/2*(sqrt(d^2 - 2*d*e)*e*f*abs(f) - (d^2 - 2*d*e)*f^2 + (d^2 - 3*d*e + 2*e ^2)*f^2)*arctan(4*sqrt(1/2)*x/sqrt(-(4*d*f - 4*e*f + sqrt(-16*e^2*f^2 + 16 *(d*f - e*f)^2))/f^2))/((sqrt(d^2 - 2*d*e)*sqrt(-d*f + 2*e*f)*e*f + sqrt(d ^2 - 2*d*e)*sqrt(-d*f)*e*f)*abs(f)) - 1/4*(sqrt(-d*f + 2*e*f)*d^2*f^2 + sq rt(-d*f)*d^2*f^2 - 3*sqrt(-d*f + 2*e*f)*d*e*f^2 - 3*sqrt(-d*f)*d*e*f^2 + 2 *sqrt(-d*f + 2*e*f)*e^2*f^2 + 2*sqrt(-d*f)*e^2*f^2 - (sqrt(-d*f + 2*e*f)*d ^2 + sqrt(-d*f)*d^2 - 2*sqrt(-d*f + 2*e*f)*d*e - 2*sqrt(-d*f)*d*e)*f^2 - ( sqrt(d^2 - 2*d*e)*sqrt(-d*f + 2*e*f)*e*f + sqrt(d^2 - 2*d*e)*sqrt(-d*f)*e* f)*abs(f))*arctan(4*sqrt(1/2)*x/sqrt(-(4*d*f - 4*e*f - sqrt(-16*e^2*f^2 + 16*(d*f - e*f)^2))/f^2))/(sqrt(d^2 - 2*d*e)*e^2*f^2*abs(f))
Time = 23.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\mathrm {atanh}\left (\frac {2\,\sqrt {d}\,\sqrt {f}\,x}{2\,f\,x^2+e}\right )}{2\,\sqrt {d}\,\sqrt {f}} \] Input:
int((e - 2*f*x^2)/(e^2 + 4*f^2*x^4 - 4*d*f*x^2 + 4*e*f*x^2),x)
Output:
atanh((2*d^(1/2)*f^(1/2)*x)/(e + 2*f*x^2))/(2*d^(1/2)*f^(1/2))
Time = 0.14 (sec) , antiderivative size = 297, normalized size of antiderivative = 7.82 \[ \int \frac {e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\sqrt {f}\, \sqrt {2}\, \left (2 \sqrt {d}\, \sqrt {d -2 e}\, \sqrt {\sqrt {d}\, \sqrt {d -2 e}-d +e}\, \mathit {atan} \left (\frac {2 f x}{\sqrt {f}\, \sqrt {\sqrt {d}\, \sqrt {d -2 e}-d +e}\, \sqrt {2}}\right )+2 \sqrt {\sqrt {d}\, \sqrt {d -2 e}-d +e}\, \mathit {atan} \left (\frac {2 f x}{\sqrt {f}\, \sqrt {\sqrt {d}\, \sqrt {d -2 e}-d +e}\, \sqrt {2}}\right ) d +\sqrt {d}\, \sqrt {d -2 e}\, \sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}\, \mathrm {log}\left (\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}-\sqrt {f}\, \sqrt {2}\, x \right )-\sqrt {d}\, \sqrt {d -2 e}\, \sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}\, \mathrm {log}\left (\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}+\sqrt {f}\, \sqrt {2}\, x \right )-\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}\, \mathrm {log}\left (\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}-\sqrt {f}\, \sqrt {2}\, x \right ) d +\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}\, \mathrm {log}\left (\sqrt {\sqrt {d}\, \sqrt {d -2 e}+d -e}+\sqrt {f}\, \sqrt {2}\, x \right ) d \right )}{8 d e f} \] Input:
int((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x)
Output:
(sqrt(f)*sqrt(2)*(2*sqrt(d)*sqrt(d - 2*e)*sqrt(sqrt(d)*sqrt(d - 2*e) - d + e)*atan((2*f*x)/(sqrt(f)*sqrt(sqrt(d)*sqrt(d - 2*e) - d + e)*sqrt(2))) + 2*sqrt(sqrt(d)*sqrt(d - 2*e) - d + e)*atan((2*f*x)/(sqrt(f)*sqrt(sqrt(d)*s qrt(d - 2*e) - d + e)*sqrt(2)))*d + sqrt(d)*sqrt(d - 2*e)*sqrt(sqrt(d)*sqr t(d - 2*e) + d - e)*log(sqrt(sqrt(d)*sqrt(d - 2*e) + d - e) - sqrt(f)*sqrt (2)*x) - sqrt(d)*sqrt(d - 2*e)*sqrt(sqrt(d)*sqrt(d - 2*e) + d - e)*log(sqr t(sqrt(d)*sqrt(d - 2*e) + d - e) + sqrt(f)*sqrt(2)*x) - sqrt(sqrt(d)*sqrt( d - 2*e) + d - e)*log(sqrt(sqrt(d)*sqrt(d - 2*e) + d - e) - sqrt(f)*sqrt(2 )*x)*d + sqrt(sqrt(d)*sqrt(d - 2*e) + d - e)*log(sqrt(sqrt(d)*sqrt(d - 2*e ) + d - e) + sqrt(f)*sqrt(2)*x)*d))/(8*d*e*f)