Integrand size = 21, antiderivative size = 75 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \] Output:
1/2*arctan(-1+2^(1/2)*e^(1/2)*x/d^(1/2))*2^(1/2)/d^(1/2)/e^(1/2)+1/2*arcta n(1+2^(1/2)*e^(1/2)*x/d^(1/2))*2^(1/2)/d^(1/2)/e^(1/2)
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {-\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )+\arctan \left (1+\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}} \] Input:
Integrate[(d + e*x^2)/(d^2 + e^2*x^4),x]
Output:
(-ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]] + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x) /Sqrt[d]])/(Sqrt[2]*Sqrt[d]*Sqrt[e])
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1476, 1082, 217}
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.
\(\displaystyle \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}+\frac {d}{e}}dx}{2 e}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {e}}+\frac {d}{e}}dx}{2 e}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} \sqrt {e}}\) |
Input:
Int[(d + e*x^2)/(d^2 + e^2*x^4),x]
Output:
-(ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*Sqrt[e])) + Arc Tan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*Sqrt[e])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11
method | result | size |
risch | \(-\frac {\sqrt {2}\, \ln \left (e \,x^{2} \sqrt {-d e}-d e x \sqrt {2}-d \sqrt {-d e}\right )}{4 \sqrt {-d e}}+\frac {\sqrt {2}\, \ln \left (e \,x^{2} \sqrt {-d e}+d e x \sqrt {2}-d \sqrt {-d e}\right )}{4 \sqrt {-d e}}\) | \(83\) |
default | \(\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d^{2}}{e^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}\) | \(232\) |
Input:
int((e*x^2+d)/(e^2*x^4+d^2),x,method=_RETURNVERBOSE)
Output:
-1/4*2^(1/2)/(-d*e)^(1/2)*ln(e*x^2*(-d*e)^(1/2)-d*e*x*2^(1/2)-d*(-d*e)^(1/ 2))+1/4*2^(1/2)/(-d*e)^(1/2)*ln(e*x^2*(-d*e)^(1/2)+d*e*x*2^(1/2)-d*(-d*e)^ (1/2))
Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.83 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {2} \sqrt {-d e} \log \left (\frac {e^{2} x^{4} - 4 \, d e x^{2} - 2 \, \sqrt {2} {\left (e x^{3} - d x\right )} \sqrt {-d e} + d^{2}}{e^{2} x^{4} + d^{2}}\right )}{4 \, d e}, \frac {\sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} \sqrt {d e} x}{2 \, d}\right ) + \sqrt {2} \sqrt {d e} \arctan \left (\frac {\sqrt {2} {\left (e x^{3} + d x\right )} \sqrt {d e}}{2 \, d^{2}}\right )}{2 \, d e}\right ] \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="fricas")
Output:
[-1/4*sqrt(2)*sqrt(-d*e)*log((e^2*x^4 - 4*d*e*x^2 - 2*sqrt(2)*(e*x^3 - d*x )*sqrt(-d*e) + d^2)/(e^2*x^4 + d^2))/(d*e), 1/2*(sqrt(2)*sqrt(d*e)*arctan( 1/2*sqrt(2)*sqrt(d*e)*x/d) + sqrt(2)*sqrt(d*e)*arctan(1/2*sqrt(2)*(e*x^3 + d*x)*sqrt(d*e)/d^2))/(d*e)]
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=- \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (- \sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {1}{d e}} \log {\left (\sqrt {2} d x \sqrt {- \frac {1}{d e}} - \frac {d}{e} + x^{2} \right )}}{4} \] Input:
integrate((e*x**2+d)/(e**2*x**4+d**2),x)
Output:
-sqrt(2)*sqrt(-1/(d*e))*log(-sqrt(2)*d*x*sqrt(-1/(d*e)) - d/e + x**2)/4 + sqrt(2)*sqrt(-1/(d*e))*log(sqrt(2)*d*x*sqrt(-1/(d*e)) - d/e + x**2)/4
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (51) = 102\).
Time = 0.11 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.03 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2} {\left (e + \sqrt {e^{2}}\right )} \log \left (\frac {2 \, \sqrt {e^{2}} x + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} - \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}{2 \, \sqrt {e^{2}} x + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}\right )}{8 \, \sqrt {e^{2}} \sqrt {-d \sqrt {e^{2}}}} + \frac {\sqrt {2} {\left (e + \sqrt {e^{2}}\right )} \log \left (\frac {2 \, \sqrt {e^{2}} x - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} - \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}{2 \, \sqrt {e^{2}} x - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-d \sqrt {e^{2}}}}\right )}{8 \, \sqrt {e^{2}} \sqrt {-d \sqrt {e^{2}}}} - \frac {\sqrt {2} {\left (e - \sqrt {e^{2}}\right )} \log \left (\sqrt {e^{2}} x^{2} + \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} x + d\right )}{8 \, \sqrt {d} {\left (e^{2}\right )}^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (e - \sqrt {e^{2}}\right )} \log \left (\sqrt {e^{2}} x^{2} - \sqrt {2} \sqrt {d} {\left (e^{2}\right )}^{\frac {1}{4}} x + d\right )}{8 \, \sqrt {d} {\left (e^{2}\right )}^{\frac {3}{4}}} \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="maxima")
Output:
1/8*sqrt(2)*(e + sqrt(e^2))*log((2*sqrt(e^2)*x + sqrt(2)*sqrt(d)*(e^2)^(1/ 4) - sqrt(2)*sqrt(-d*sqrt(e^2)))/(2*sqrt(e^2)*x + sqrt(2)*sqrt(d)*(e^2)^(1 /4) + sqrt(2)*sqrt(-d*sqrt(e^2))))/(sqrt(e^2)*sqrt(-d*sqrt(e^2))) + 1/8*sq rt(2)*(e + sqrt(e^2))*log((2*sqrt(e^2)*x - sqrt(2)*sqrt(d)*(e^2)^(1/4) - s qrt(2)*sqrt(-d*sqrt(e^2)))/(2*sqrt(e^2)*x - sqrt(2)*sqrt(d)*(e^2)^(1/4) + sqrt(2)*sqrt(-d*sqrt(e^2))))/(sqrt(e^2)*sqrt(-d*sqrt(e^2))) - 1/8*sqrt(2)* (e - sqrt(e^2))*log(sqrt(e^2)*x^2 + sqrt(2)*sqrt(d)*(e^2)^(1/4)*x + d)/(sq rt(d)*(e^2)^(3/4)) + 1/8*sqrt(2)*(e - sqrt(e^2))*log(sqrt(e^2)*x^2 - sqrt( 2)*sqrt(d)*(e^2)^(1/4)*x + d)/(sqrt(d)*(e^2)^(3/4))
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2} \sqrt {-d e} \log \left (x^{2} + \sqrt {2} x \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{e^{2}}}\right )}{4 \, d e} - \frac {\sqrt {2} \sqrt {-d e} \log \left (x^{2} - \sqrt {2} x \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{e^{2}}}\right )}{4 \, d e} \] Input:
integrate((e*x^2+d)/(e^2*x^4+d^2),x, algorithm="giac")
Output:
1/4*sqrt(2)*sqrt(-d*e)*log(x^2 + sqrt(2)*x*(d^2/e^2)^(1/4) + sqrt(d^2/e^2) )/(d*e) - 1/4*sqrt(2)*sqrt(-d*e)*log(x^2 - sqrt(2)*x*(d^2/e^2)^(1/4) + sqr t(d^2/e^2))/(d*e)
Time = 17.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,e^{3/2}\,x^3}{2\,d^{3/2}}+\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )\right )}{4\,\sqrt {d}\,\sqrt {e}} \] Input:
int((d + e*x^2)/(d^2 + e^2*x^4),x)
Output:
(2^(1/2)*(2*atan((2^(1/2)*e^(1/2)*x)/(2*d^(1/2))) + 2*atan((2^(1/2)*e^(3/2 )*x^3)/(2*d^(3/2)) + (2^(1/2)*e^(1/2)*x)/(2*d^(1/2)))))/(4*d^(1/2)*e^(1/2) )
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {2}\, \left (-\mathit {atan} \left (\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {2}-2 e x}{\sqrt {e}\, \sqrt {d}\, \sqrt {2}}\right )+\mathit {atan} \left (\frac {\sqrt {e}\, \sqrt {d}\, \sqrt {2}+2 e x}{\sqrt {e}\, \sqrt {d}\, \sqrt {2}}\right )\right )}{2 d e} \] Input:
int((e*x^2+d)/(e^2*x^4+d^2),x)
Output:
(sqrt(e)*sqrt(d)*sqrt(2)*( - atan((sqrt(e)*sqrt(d)*sqrt(2) - 2*e*x)/(sqrt( e)*sqrt(d)*sqrt(2))) + atan((sqrt(e)*sqrt(d)*sqrt(2) + 2*e*x)/(sqrt(e)*sqr t(d)*sqrt(2)))))/(2*d*e)