\(\int \frac {\sqrt {d-e x^2}}{(d^2-e^2 x^4)^{3/2}} \, dx\) [161]

Optimal result

Integrand size = 29, antiderivative size = 95 \[ \int \frac {\sqrt {d-e x^2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \sqrt {d-e x^2}}{2 d^2 \sqrt {d^2-e^2 x^4}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 3.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {d-e x^2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d+e x^2}+\sqrt {2} \left (d+e x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{4 d^2 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[Sqrt[d - e*x^2]/(d^2 - e^2*x^4)^(3/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d + e*x^2] + Sqrt[2]*(d + e*x^2)*Ar 
cTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]))/(4*d^2*Sqrt[e]*Sqrt[d - e*x^2 
]*(d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1396, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[d - e*x^2]/(d^2 - e^2*x^4)^(3/2),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(2*d^2*Sqrt[d + e*x^2]) + ArcTanh[(Sqr 
t[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*Sqrt[2]*d^2*Sqrt[e])))/Sqrt[d^2 - e^2* 
x^4]
 

Defintions of rubi rules used

rule 221

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]
 

rule 291

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]
 

rule 296

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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(75)=150\).

Time = 0.80 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.61

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} \left (\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, d e \,x^{2}-\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, d e \,x^{2}+4 \sqrt {d e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\, \sqrt {d}\, x +\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, d^{2}-\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, d^{2}\right )}{4 \sqrt {-e \,x^{2}+d}\, \sqrt {e \,x^{2}+d}\, \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, d^{\frac {3}{2}} \left (e x -\sqrt {-d e}\right ) \left (e x +\sqrt {-d e}\right )}\)	\(343\)

Input:

int((-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(-e^2*x^4+d^2)^(1/2)*e^2*(ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)+(d* 
e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*2^(1/2)*d*e*x^2-ln(2*e*(2^(1/2)*d^(1/2)*( 
e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2)))*2^(1/2)*d*e*x^2+4*(d*e) 
^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)*d^(1/2)*x+ln(2*e* 
(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*2^(1/ 
2)*d^2-ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e) 
^(1/2)))*2^(1/2)*d^2)/(-e*x^2+d)^(1/2)/(e*x^2+d)^(1/2)/(-(d*e)^(1/2)+(-d*e 
)^(1/2))/((d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)/d^(3/2)/(e*x-(-d*e)^(1/2)) 
/(e*x+(-d*e)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt {d-e x^2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} e x - \sqrt {2} {\left (e^{2} x^{4} - d^{2}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, {\left (d^{2} e^{3} x^{4} - d^{4} e\right )}}, -\frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} e x - \sqrt {2} {\left (e^{2} x^{4} - d^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right )}{4 \, {\left (d^{2} e^{3} x^{4} - d^{4} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[-1/8*(4*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*e*x - sqrt(2)*(e^2*x^4 - d^ 
2)*sqrt(e)*log(-(3*e^2*x^4 - 2*d*e*x^2 - 2*sqrt(2)*sqrt(-e^2*x^4 + d^2)*sq 
rt(-e*x^2 + d)*sqrt(e)*x - d^2)/(e^2*x^4 - 2*d*e*x^2 + d^2)))/(d^2*e^3*x^4 
 - d^4*e), -1/4*(2*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*e*x - sqrt(2)*(e^ 
2*x^4 - d^2)*sqrt(-e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d) 
*sqrt(-e)*x/(e^2*x^4 - d^2)))/(d^2*e^3*x^4 - d^4*e)]
 

Sympy [F]

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

integrate((-e*x**2+d)**(1/2)/(-e**2*x**4+d**2)**(3/2),x)
 

Output:

Integral(sqrt(d - e*x**2)/(-(-d + e*x**2)*(d + e*x**2))**(3/2), x)
 

Maxima [F]

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

integrate((-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-e*x^2 + d)/(-e^2*x^4 + d^2)^(3/2), x)
 

Giac [F]

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

integrate((-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-e*x^2 + d)/(-e^2*x^4 + d^2)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d-e x^2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {d-e\,x^2}}{{\left (d^2-e^2\,x^4\right )}^{3/2}} \,d x \] Input:

int((d - e*x^2)^(1/2)/(d^2 - e^2*x^4)^(3/2),x)
 

Output:

int((d - e*x^2)^(1/2)/(d^2 - e^2*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt {d-e x^2}}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {e \,x^{2}+d}\, e x -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d +\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d +\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}-\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d -\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e \,x^{2}+4 \sqrt {e}\, d +4 \sqrt {e}\, e \,x^{2}}{8 d^{2} e \left (e \,x^{2}+d \right )} \] Input:

int((-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),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))