\(\int \frac {d-e x^2}{(d+e x^2) \sqrt {d^2-e^2 x^4}} \, dx\) [83]

Optimal result

Integrand size = 34, antiderivative size = 72 \[ \int \frac {d-e x^2}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {\left (d-e x^2\right ) E\left (\left .\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |2\right )}{\sqrt {d} \sqrt {e} \sqrt {\frac {d-e x^2}{d+e x^2}} \sqrt {d^2-e^2 x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int \frac {d-e x^2}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {x-\frac {e x^3}{d}-\frac {i \sqrt {1-\frac {e^2 x^4}{d^2}} \left (E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\sqrt {-\frac {e}{d}}}}{\sqrt {d^2-e^2 x^4}} \] Input:

Integrate[(d - e*x^2)/((d + e*x^2)*Sqrt[d^2 - e^2*x^4]),x]
 

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(72)=144\).

Time = 0.75 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.24, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1396, 314, 25, 27, 344, 836, 27, 765, 762, 1390, 1389, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[(d - e*x^2)/((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])/(d*Sqrt[d + e*x^2]) 
+ (e*Sqrt[d^2 - e^2*x^4]*((d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSi 
n[(Sqrt[e]*x)/Sqrt[d]], -1])/(e^(3/2)*Sqrt[d^2 - e^2*x^4]) - (d^(3/2)*Sqrt 
[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(e^(3/2)*S 
qrt[d^2 - e^2*x^4])))/(d*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/Sqrt[d^2 - e^2 
*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

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 344

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

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 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 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 [C] (verified)

Result contains complex when optimal does not.

Time = 2.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.68

method	result	size
elliptic	\(\frac {\left (-e^{2} x^{2}+d e \right ) x}{d e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(121\)
default	\(-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+2 d \left (\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )\)	\(252\)

Input:

int((-e*x^2+d)/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(-e^2*x^2+d*e)/d*x/e/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)-1/(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 = 85, normalized size of antiderivative = 1.18 \[ \int \frac {d-e x^2}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\frac {{\left (e x^{2} + d\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left (e x^{2} + d\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} x}{d e x^{2} + d^{2}} \] Input:

integrate((-e*x^2+d)/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
 

Output:

((e*x^2 + d)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) - (e*x^2 + d)*s 
qrt(e/d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + sqrt(-e^2*x^4 + d^2)*x)/(d* 
e*x^2 + d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {d-e x^2}{\left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-e*x^2+d)/(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]

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

integrate((-e*x^2+d)/(e*x^2+d)/(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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