Integrand size = 26, antiderivative size = 61 \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \] Output:
1/2*x/d^2/(e*x^2+d)^(1/2)+1/4*arctanh(2^(1/2)*e^(1/2)*x/(e*x^2+d)^(1/2))*2 ^(1/2)/d^2/e^(1/2)
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\frac {\frac {2 x}{\sqrt {d+e x^2}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {e}}}{4 d^2} \] Input:
Integrate[1/(Sqrt[d + e*x^2]*(d^2 - e^2*x^4)),x]
Output:
((2*x)/Sqrt[d + e*x^2] + (Sqrt[2]*ArcTanh[(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2])/(Sqrt[2]*d)])/Sqrt[e])/(4*d^2)
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1388, 296, 291, 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.
\(\displaystyle \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 296 |
\(\displaystyle \frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d}+\frac {x}{2 d^2 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 d}+\frac {x}{2 d^2 \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}}+\frac {x}{2 d^2 \sqrt {d+e x^2}}\) |
Input:
Int[1/(Sqrt[d + e*x^2]*(d^2 - e^2*x^4)),x]
Output:
x/(2*d^2*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/( 2*Sqrt[2]*d^2*Sqrt[e])
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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
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 = 0.85 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2}}{2 x \sqrt {e}}\right )}{2 \sqrt {e}}+\frac {x}{\sqrt {e \,x^{2}+d}}}{2 d^{2}}\) | \(47\) |
default | \(-\frac {e \sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}{x -\frac {\sqrt {d e}}{e}}\right )}{4 \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, \sqrt {d}}+\frac {e \sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}{x +\frac {\sqrt {d e}}{e}}\right )}{4 \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, \sqrt {d}}-\frac {\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{2 d \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right )}-\frac {\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{2 d \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right )}\) | \(441\) |
Input:
int(1/(e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)
Output:
1/2*(1/2*2^(1/2)/e^(1/2)*arctanh(1/2*(e*x^2+d)^(1/2)/x*2^(1/2)/e^(1/2))+1/ (e*x^2+d)^(1/2)*x)/d^2
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (45) = 90\).
Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.43 \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\left [\frac {\sqrt {2} {\left (e x^{2} + d\right )} \sqrt {e} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 8 \, \sqrt {e x^{2} + d} e x}{16 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}}, -\frac {\sqrt {2} {\left (e x^{2} + d\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right ) - 4 \, \sqrt {e x^{2} + d} e x}{8 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}}\right ] \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="fricas")
Output:
[1/16*(sqrt(2)*(e*x^2 + d)*sqrt(e)*log((17*e^2*x^4 + 14*d*e*x^2 + 4*sqrt(2 )*(3*e*x^3 + d*x)*sqrt(e*x^2 + d)*sqrt(e) + d^2)/(e^2*x^4 - 2*d*e*x^2 + d^ 2)) + 8*sqrt(e*x^2 + d)*e*x)/(d^2*e^2*x^2 + d^3*e), -1/8*(sqrt(2)*(e*x^2 + d)*sqrt(-e)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(e*x^2 + d)*sqrt(-e)/(e^ 2*x^3 + d*e*x)) - 4*sqrt(e*x^2 + d)*e*x)/(d^2*e^2*x^2 + d^3*e)]
\[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=- \int \frac {1}{- d^{2} \sqrt {d + e x^{2}} + e^{2} x^{4} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate(1/(e*x**2+d)**(1/2)/(-e**2*x**4+d**2),x)
Output:
-Integral(1/(-d**2*sqrt(d + e*x**2) + e**2*x**4*sqrt(d + e*x**2)), x)
\[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\int { -\frac {1}{{\left (e^{2} x^{4} - d^{2}\right )} \sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="maxima")
Output:
-integrate(1/((e^2*x^4 - d^2)*sqrt(e*x^2 + d)), x)
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (45) = 90\).
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.66 \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{8 \, d \sqrt {e} {\left | d \right |}} + \frac {x}{2 \, \sqrt {e x^{2} + d} d^{2}} \] Input:
integrate(1/(e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="giac")
Output:
1/8*sqrt(2)*log(abs(2*(sqrt(e)*x - sqrt(e*x^2 + d))^2 - 4*sqrt(2)*abs(d) - 6*d)/abs(2*(sqrt(e)*x - sqrt(e*x^2 + d))^2 + 4*sqrt(2)*abs(d) - 6*d))/(d* sqrt(e)*abs(d)) + 1/2*x/(sqrt(e*x^2 + d)*d^2)
Timed out. \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\int \frac {1}{\left (d^2-e^2\,x^4\right )\,\sqrt {e\,x^2+d}} \,d x \] Input:
int(1/((d^2 - e^2*x^4)*(d + e*x^2)^(1/2)),x)
Output:
int(1/((d^2 - e^2*x^4)*(d + e*x^2)^(1/2)), x)
Time = 0.18 (sec) , antiderivative size = 327, normalized size of antiderivative = 5.36 \[ \int \frac {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx=\frac {4 \sqrt {e \,x^{2}+d}\, e x -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d +\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d +\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}-\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+4 \sqrt {e}\, d +4 \sqrt {e}\, e \,x^{2}}{8 d^{2} e \left (e \,x^{2}+d \right )} \] Input:
int(1/(e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x)
Output:
(4*sqrt(d + e*x**2)*e*x - sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)* sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d - sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e*x**2 + sqrt(e) *sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sq rt(d))*d + sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt( d) + sqrt(e)*x)/sqrt(d))*e*x**2 + sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d + sqrt(e)*sqrt(2)*log((s qrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e*x**2 - sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt( e)*x)/sqrt(d))*d - sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*e*x**2 + 4*sqrt(e)*d + 4*sqrt(e)*e*x**2)/ (8*d**2*e*(d + e*x**2))