Integrand size = 24, antiderivative size = 72 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}} \] Output:
1/4*x/d^2/(e*x^2+d)+1/2*arctan(e^(1/2)*x/d^(1/2))/d^(5/2)/e^(1/2)+1/4*arct anh(e^(1/2)*x/d^(1/2))/d^(5/2)/e^(1/2)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {\frac {\sqrt {d} x}{d+e x^2}+\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}}{4 d^{5/2}} \] Input:
Integrate[1/((d + e*x^2)*(d^2 - e^2*x^4)),x]
Output:
((Sqrt[d]*x)/(d + e*x^2) + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + ArcTa nh[(Sqrt[e]*x)/Sqrt[d]]/Sqrt[e])/(4*d^(5/2))
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1388, 316, 25, 27, 397, 218, 221}
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.
\(\displaystyle \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^2}dx\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {x}{4 d^2 \left (d+e x^2\right )}-\frac {\int -\frac {e \left (3 d-e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )}dx}{4 d^2 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {e \left (3 d-e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )}dx}{4 d^2 e}+\frac {x}{4 d^2 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {3 d-e x^2}{\left (d-e x^2\right ) \left (e x^2+d\right )}dx}{4 d^2}+\frac {x}{4 d^2 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {\int \frac {1}{d-e x^2}dx+2 \int \frac {1}{e x^2+d}dx}{4 d^2}+\frac {x}{4 d^2 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\int \frac {1}{d-e x^2}dx+\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}}{4 d^2}+\frac {x}{4 d^2 \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}}}{4 d^2}+\frac {x}{4 d^2 \left (d+e x^2\right )}\) |
Input:
Int[1/((d + e*x^2)*(d^2 - e^2*x^4)),x]
Output:
x/(4*d^2*(d + e*x^2)) + ((2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e]))/(4*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {x}{e \,x^{2}+d}+\frac {2 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}}{4 d^{2}}+\frac {\operatorname {arctanh}\left (\frac {e x}{\sqrt {d e}}\right )}{4 d^{2} \sqrt {d e}}\) | \(54\) |
risch | \(\frac {x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {\ln \left (-e x -\sqrt {-d e}\right )}{4 \sqrt {-d e}\, d^{2}}+\frac {\ln \left (e x -\sqrt {-d e}\right )}{4 \sqrt {-d e}\, d^{2}}+\frac {\ln \left (e x +\sqrt {d e}\right )}{8 \sqrt {d e}\, d^{2}}-\frac {\ln \left (-e x +\sqrt {d e}\right )}{8 \sqrt {d e}\, d^{2}}\) | \(107\) |
Input:
int(1/(e*x^2+d)/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)
Output:
1/4/d^2*(x/(e*x^2+d)+2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))+1/4/d^2/(d*e)^ (1/2)*arctanh(e*x/(d*e)^(1/2))
Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.62 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\left [\frac {2 \, d e x + 4 \, {\left (e x^{2} + d\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (e x^{2} + d\right )} \sqrt {d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {d e} x + d}{e x^{2} - d}\right )}{8 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {d e x - {\left (e x^{2} + d\right )} \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right ) - {\left (e x^{2} + d\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{4 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="fricas")
Output:
[1/8*(2*d*e*x + 4*(e*x^2 + d)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (e*x^2 + d )*sqrt(d*e)*log((e*x^2 + 2*sqrt(d*e)*x + d)/(e*x^2 - d)))/(d^3*e^2*x^2 + d ^4*e), 1/4*(d*e*x - (e*x^2 + d)*sqrt(-d*e)*arctan(sqrt(-d*e)*x/d) - (e*x^2 + d)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)))/(d^3*e^2*x ^2 + d^4*e)]
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (63) = 126\).
Time = 0.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.14 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4 d^{3} + 4 d^{2} e x^{2}} - \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (- \frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} - \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} + \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (\frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} + \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} - \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (- \frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} - \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (\frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} + \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} \] Input:
integrate(1/(e*x**2+d)/(-e**2*x**4+d**2),x)
Output:
x/(4*d**3 + 4*d**2*e*x**2) - sqrt(1/(d**5*e))*log(-d**8*e*(1/(d**5*e))**(3 /2)/10 - 9*d**3*sqrt(1/(d**5*e))/10 + x)/8 + sqrt(1/(d**5*e))*log(d**8*e*( 1/(d**5*e))**(3/2)/10 + 9*d**3*sqrt(1/(d**5*e))/10 + x)/8 - sqrt(-1/(d**5* e))*log(-4*d**8*e*(-1/(d**5*e))**(3/2)/5 - 9*d**3*sqrt(-1/(d**5*e))/5 + x) /4 + sqrt(-1/(d**5*e))*log(4*d**8*e*(-1/(d**5*e))**(3/2)/5 + 9*d**3*sqrt(- 1/(d**5*e))/5 + x)/4
Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{2}} - \frac {\arctan \left (\frac {e x}{\sqrt {-d e}}\right )}{4 \, \sqrt {-d e} d^{2}} + \frac {x}{4 \, {\left (e x^{2} + d\right )} d^{2}} \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="giac")
Output:
1/2*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^2) - 1/4*arctan(e*x/sqrt(-d*e))/(sq rt(-d*e)*d^2) + 1/4*x/((e*x^2 + d)*d^2)
Time = 17.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4\,d^2\,\left (e\,x^2+d\right )}+\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {d^5\,e}}{d^3}\right )\,\sqrt {d^5\,e}}{4\,d^5\,e}-\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {-d^5\,e}}{d^3}\right )\,\sqrt {-d^5\,e}}{2\,d^5\,e} \] Input:
int(1/((d^2 - e^2*x^4)*(d + e*x^2)),x)
Output:
x/(4*d^2*(d + e*x^2)) + (atanh((x*(d^5*e)^(1/2))/d^3)*(d^5*e)^(1/2))/(4*d^ 5*e) - (atanh((x*(-d^5*e)^(1/2))/d^3)*(-d^5*e)^(1/2))/(2*d^5*e)
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d +4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e \,x^{2}+\sqrt {e}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {e}\, \sqrt {d}-e x \right ) d +\sqrt {e}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {e}\, \sqrt {d}-e x \right ) e \,x^{2}-\sqrt {e}\, \sqrt {d}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {d}-e x \right ) d -\sqrt {e}\, \sqrt {d}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {d}-e x \right ) e \,x^{2}+2 d e x}{8 d^{3} e \left (e \,x^{2}+d \right )} \] Input:
int(1/(e*x^2+d)/(-e^2*x^4+d^2),x)
Output:
(4*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d + 4*sqrt(e)*sqrt(d)*ata n((e*x)/(sqrt(e)*sqrt(d)))*e*x**2 + sqrt(e)*sqrt(d)*log( - sqrt(e)*sqrt(d) - e*x)*d + sqrt(e)*sqrt(d)*log( - sqrt(e)*sqrt(d) - e*x)*e*x**2 - sqrt(e) *sqrt(d)*log(sqrt(e)*sqrt(d) - e*x)*d - sqrt(e)*sqrt(d)*log(sqrt(e)*sqrt(d ) - e*x)*e*x**2 + 2*d*e*x)/(8*d**3*e*(d + e*x**2))