\(\int \frac {1}{(d-e x^2) \sqrt {d^2-e^2 x^4}} \, dx\) [108]

Optimal result

Integrand size = 27, antiderivative size = 156 \[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{2 d^2 \left (d-e x^2\right )}-\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] 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)*Ellipt 
icE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+(1-e^2*x^4/d 
^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1 
/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (d+e x^2\right )+i d \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 i d \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{2 d^2 \sqrt {-\frac {e}{d}} \sqrt {d^2-e^2 x^4}} \] 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*I)*d*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*Arc 
Sinh[Sqrt[-(e/d)]*x], -1])/(2*d^2*Sqrt[-(e/d)]*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1396, 316, 27, 326, 289, 329, 327, 765, 762}

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

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \sqrt {d-e x^2}}{\sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {\sqrt {d-e x^2}}{\sqrt {e x^2+d}}dx}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 326

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-\int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d^2}+\frac {x \sqrt {d+e x^2}}{2 d^2 \sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

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]) + (-((d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqr 
t[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])) + (2*d^(3/2)*Sqrt[1 
 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqr 
t[d - e*x^2]*Sqrt[d + e*x^2]))/(2*d^2)))/Sqrt[d^2 - e^2*x^4]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 316

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]
 

rule 326

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
b/d   Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d   In 
t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && 
PosQ[d/c] && NegQ[b/a]
 

rule 327

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]
 

rule 329

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])
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1396

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])
 

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22

method	result	size
default	\(-\frac {\left (-e^{2} x^{2}-d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(191\)
elliptic	\(-\frac {\left (-e^{2} x^{2}-d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(191\)

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))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=-\frac {\sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{2} - d e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (d e + e^{2}\right )} x^{2} - d^{2} - d e\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{2 \, {\left (d^{2} e^{2} x^{2} - d^{3} e\right )}} \] 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(arcs 
in(x*sqrt(e/d)), -1) - ((d*e + e^2)*x^2 - d^2 - d*e)*sqrt(e/d)*elliptic_f( 
arcsin(x*sqrt(e/d)), -1))/(d^2*e^2*x^2 - d^3*e)
 

Sympy [F]

\[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=- \int \frac {1}{- d \sqrt {d^{2} - e^{2} x^{4}} + e x^{2} \sqrt {d^{2} - e^{2} x^{4}}}\, dx \] Input:

integrate(1/(-e*x**2+d)/(-e**2*x**4+d**2)**(1/2),x)
 

Output:

-Integral(1/(-d*sqrt(d**2 - e**2*x**4) + e*x**2*sqrt(d**2 - e**2*x**4)), x 
)
 

Maxima [F(-2)]

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
 

Giac [F]

\[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int { -\frac {1}{\sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} - d\right )}} \,d x } \] 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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {1}{\sqrt {d^2-e^2\,x^4}\,\left (d-e\,x^2\right )} \,d x \] 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)
 

Reduce [F]

\[ \int \frac {1}{\left (d-e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{3} x^{6}-d \,e^{2} x^{4}-d^{2} e \,x^{2}+d^{3}}d x \] 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)