Integrand size = 26, antiderivative size = 96 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{2 d^2 \left (d+e x^2\right )}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/2*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)+1/2*(1-e^2*x^4/d^2)^(1/2)*Ellipti cE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (d-e x^2\right )-i d \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )}{2 d^2 \sqrt {-\frac {e}{d}} \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[1/((d + e*x^2)*Sqrt[d^2 - e^2*x^4]),x]
Output:
(Sqrt[-(e/d)]*x*(d - e*x^2) - I*d*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcS inh[Sqrt[-(e/d)]*x], -1])/(2*d^2*Sqrt[-(e/d)]*Sqrt[d^2 - e^2*x^4])
Time = 0.43 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1396, 316, 25, 27, 329, 327}
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.
\(\displaystyle \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d+e x^2}}-\frac {\int -\frac {e \sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d^2 e}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d^2 e}+\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d^2}+\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 329 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{2 d \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[1/((d + e*x^2)*Sqrt[d^2 - e^2*x^4]),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((x*Sqrt[d - e*x^2])/(2*d^2*Sqrt[d + e*x^ 2]) + (Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1]) /(2*Sqrt[d]*Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/Sqrt[d^2 - e^2*x^4]
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)^(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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 0])
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (78 ) = 156\).
Time = 1.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(188\) |
elliptic | \(\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(188\) |
Input:
int(1/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(-e^2*x^2+d*e)/d^2*x/e/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/2/d/(e/d)^(1 /2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*( e/d)^(1/2),I)-1/2/d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2* x^4+d^2)^(1/2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{2} + d e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (d e - e^{2}\right )} x^{2} + d^{2} - d e\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{2 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}} \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
Output:
1/2*(sqrt(-e^2*x^4 + d^2)*e*x + (e^2*x^2 + d*e)*sqrt(e/d)*elliptic_e(arcsi n(x*sqrt(e/d)), -1) + ((d*e - e^2)*x^2 + d^2 - d*e)*sqrt(e/d)*elliptic_f(a rcsin(x*sqrt(e/d)), -1))/(d^2*e^2*x^2 + d^3*e)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )} \left (d + e x^{2}\right )}\, dx \] Input:
integrate(1/(e*x**2+d)/(-e**2*x**4+d**2)**(1/2),x)
Output:
Integral(1/(sqrt(-(-d + e*x**2)*(d + e*x**2))*(d + e*x**2)), x)
Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),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
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-e^2*x^4 + d^2)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {1}{\sqrt {d^2-e^2\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:
int(1/((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)),x)
Output:
int(1/((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)), x)
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \] Input:
int(1/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6), x)