Integrand size = 26, antiderivative size = 38 \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}} \] Output:
1/2*arctanh(2^(1/2)*e^(1/2)*x/(e*x^2+d)^(1/2))*2^(1/2)/d/e^(1/2)
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\frac {\text {arctanh}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {2} d \sqrt {e}} \] Input:
Integrate[Sqrt[d + e*x^2]/(d^2 - e^2*x^4),x]
Output:
ArcTanh[(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2])/(Sqrt[2]*d)]/(Sqrt[2]*d*Sq rt[e])
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1388, 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.
\(\displaystyle \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}}dx\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \int \frac {1}{d-\frac {2 d e x^2}{d+e x^2}}d\frac {x}{\sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}}\) |
Input:
Int[Sqrt[d + e*x^2]/(d^2 - e^2*x^4),x]
Output:
ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(Sqrt[2]*d*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[(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 = 0.84 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2}}{2 x \sqrt {e}}\right )}{2 d \sqrt {e}}\) | \(33\) |
default | \(-\frac {e \left (\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}+\frac {\sqrt {d e}\, \ln \left (\frac {\sqrt {d e}+e \left (x -\frac {\sqrt {d e}}{e}\right )}{\sqrt {e}}+\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}\right )}{\sqrt {e}}-\sqrt {d}\, \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 )\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (\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}-\frac {\sqrt {d e}\, \ln \left (\frac {-\sqrt {d e}+e \left (x +\frac {\sqrt {d e}}{e}\right )}{\sqrt {e}}+\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}\right )}{\sqrt {e}}-\sqrt {d}\, \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 )\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}+\frac {\sqrt {-d e}\, \ln \left (\frac {\sqrt {-d e}+e \left (x -\frac {\sqrt {-d e}}{e}\right )}{\sqrt {e}}+\sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\right )}{\sqrt {e}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}-\frac {e \left (\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}-\frac {\sqrt {-d e}\, \ln \left (\frac {-\sqrt {-d e}+e \left (x +\frac {\sqrt {-d e}}{e}\right )}{\sqrt {e}}+\sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\right )}{\sqrt {e}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}\) | \(818\) |
Input:
int((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)
Output:
1/2/d*2^(1/2)/e^(1/2)*arctanh(1/2*(e*x^2+d)^(1/2)/x*2^(1/2)/e^(1/2))
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.63 \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\left [\frac {\sqrt {2} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, d \sqrt {e}}, -\frac {\sqrt {2} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right )}{4 \, d e}\right ] \] Input:
integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="fricas")
Output:
[1/8*sqrt(2)*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))/(d*sqrt(e)), -1/4* sqrt(2)*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))/(d*e)]
\[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=- \int \frac {1}{- d \sqrt {d + e x^{2}} + e x^{2} \sqrt {d + e x^{2}}}\, dx \] Input:
integrate((e*x**2+d)**(1/2)/(-e**2*x**4+d**2),x)
Output:
-Integral(1/(-d*sqrt(d + e*x**2) + e*x**2*sqrt(d + e*x**2)), x)
\[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\int { -\frac {\sqrt {e x^{2} + d}}{e^{2} x^{4} - d^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="maxima")
Output:
-integrate(sqrt(e*x^2 + d)/(e^2*x^4 - d^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (29) = 58\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{4 \, \sqrt {e} {\left | d \right |}} \] Input:
integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="giac")
Output:
1/4*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))/(sq rt(e)*abs(d))
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\int \frac {\sqrt {e\,x^2+d}}{d^2-e^2\,x^4} \,d x \] Input:
int((d + e*x^2)^(1/2)/(d^2 - e^2*x^4),x)
Output:
int((d + e*x^2)^(1/2)/(d^2 - e^2*x^4), x)
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.34 \[ \int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx=\frac {\sqrt {e}\, \sqrt {2}\, \left (-\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )+\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )+\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )-\mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right )\right )}{4 d e} \] Input:
int((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x)
Output:
(sqrt(e)*sqrt(2)*( - log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + s qrt(e)*x)/sqrt(d)) + log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + s qrt(e)*x)/sqrt(d)) + log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + s qrt(e)*x)/sqrt(d)) - log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + s qrt(e)*x)/sqrt(d))))/(4*d*e)