\(\int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} (a+c x^2)^2} \, dx\) [7]

Optimal result

Integrand size = 29, antiderivative size = 103 \[ \int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\frac {c x \sqrt {1-e^2 x^2}}{2 a \left (c+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c+2 a e^2\right ) \arctan \left (\frac {\sqrt {c+a e^2} x}{\sqrt {a} \sqrt {1-e^2 x^2}}\right )}{2 a^{3/2} \left (c+a e^2\right )^{3/2}} \] Output:

1/2*c*x*(-e^2*x^2+1)^(1/2)/a/(a*e^2+c)/(c*x^2+a)+1/2*(2*a*e^2+c)*arctan((a 
*e^2+c)^(1/2)*x/a^(1/2)/(-e^2*x^2+1)^(1/2))/a^(3/2)/(a*e^2+c)^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(926\) vs. \(2(103)=206\).

Time = 6.46 (sec) , antiderivative size = 926, normalized size of antiderivative = 8.99 \[ \int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} c x}{\left (c+a e^2\right ) \left (a+c x^2\right ) \left (-2+e^2 x^2+2 \sqrt {1-e^2 x^2}\right )}-\frac {2 \sqrt {a} c e^2 x^3}{\left (c+a e^2\right ) \left (a+c x^2\right ) \left (-2+e^2 x^2+2 \sqrt {1-e^2 x^2}\right )}-\frac {2 \sqrt {a} c x \sqrt {1-e^2 x^2}}{\left (c+a e^2\right ) \left (a+c x^2\right ) \left (-2+e^2 x^2+2 \sqrt {1-e^2 x^2}\right )}+\frac {\sqrt {a} c e^2 x^3 \sqrt {1-e^2 x^2}}{\left (c+a e^2\right ) \left (a+c x^2\right ) \left (-2+e^2 x^2+2 \sqrt {1-e^2 x^2}\right )}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {2 c+a e^2-2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (1-\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right )^{3/2} \sqrt {2 c+a e^2-2 \sqrt {c} \sqrt {c+a e^2}}}+\frac {\left (c \sqrt {c+a e^2}+2 a e^2 \left (-\sqrt {c}+\sqrt {c+a e^2}\right )\right ) \arctan \left (\frac {\sqrt {2 c+a e^2-2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (-1+\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right )^{3/2} \sqrt {2 c+a e^2-2 \sqrt {c} \sqrt {c+a e^2}}}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (-1+\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right )^{3/2} \sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}}}+\frac {2 a \sqrt {c} e^2 \arctan \left (\frac {\sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (-1+\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right )^{3/2} \sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}}}+\frac {c \arctan \left (\frac {\sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (-1+\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right ) \sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}}}+\frac {2 a e^2 \arctan \left (\frac {\sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}} x}{\sqrt {a} \left (-1+\sqrt {1-e^2 x^2}\right )}\right )}{\left (c+a e^2\right ) \sqrt {2 c+a e^2+2 \sqrt {c} \sqrt {c+a e^2}}}}{2 a^{3/2}} \] Input:

Integrate[1/(Sqrt[1 - e*x]*Sqrt[1 + e*x]*(a + c*x^2)^2),x]
 

Output:

((2*Sqrt[a]*c*x)/((c + a*e^2)*(a + c*x^2)*(-2 + e^2*x^2 + 2*Sqrt[1 - e^2*x 
^2])) - (2*Sqrt[a]*c*e^2*x^3)/((c + a*e^2)*(a + c*x^2)*(-2 + e^2*x^2 + 2*S 
qrt[1 - e^2*x^2])) - (2*Sqrt[a]*c*x*Sqrt[1 - e^2*x^2])/((c + a*e^2)*(a + c 
*x^2)*(-2 + e^2*x^2 + 2*Sqrt[1 - e^2*x^2])) + (Sqrt[a]*c*e^2*x^3*Sqrt[1 - 
e^2*x^2])/((c + a*e^2)*(a + c*x^2)*(-2 + e^2*x^2 + 2*Sqrt[1 - e^2*x^2])) + 
 (c^(3/2)*ArcTan[(Sqrt[2*c + a*e^2 - 2*Sqrt[c]*Sqrt[c + a*e^2]]*x)/(Sqrt[a 
]*(1 - Sqrt[1 - e^2*x^2]))])/((c + a*e^2)^(3/2)*Sqrt[2*c + a*e^2 - 2*Sqrt[ 
c]*Sqrt[c + a*e^2]]) + ((c*Sqrt[c + a*e^2] + 2*a*e^2*(-Sqrt[c] + Sqrt[c + 
a*e^2]))*ArcTan[(Sqrt[2*c + a*e^2 - 2*Sqrt[c]*Sqrt[c + a*e^2]]*x)/(Sqrt[a] 
*(-1 + Sqrt[1 - e^2*x^2]))])/((c + a*e^2)^(3/2)*Sqrt[2*c + a*e^2 - 2*Sqrt[ 
c]*Sqrt[c + a*e^2]]) + (c^(3/2)*ArcTan[(Sqrt[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[ 
c + a*e^2]]*x)/(Sqrt[a]*(-1 + Sqrt[1 - e^2*x^2]))])/((c + a*e^2)^(3/2)*Sqr 
t[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c + a*e^2]]) + (2*a*Sqrt[c]*e^2*ArcTan[(Sqr 
t[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c + a*e^2]]*x)/(Sqrt[a]*(-1 + Sqrt[1 - e^2* 
x^2]))])/((c + a*e^2)^(3/2)*Sqrt[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c + a*e^2]]) 
 + (c*ArcTan[(Sqrt[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c + a*e^2]]*x)/(Sqrt[a]*(- 
1 + Sqrt[1 - e^2*x^2]))])/((c + a*e^2)*Sqrt[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c 
 + a*e^2]]) + (2*a*e^2*ArcTan[(Sqrt[2*c + a*e^2 + 2*Sqrt[c]*Sqrt[c + a*e^2 
]]*x)/(Sqrt[a]*(-1 + Sqrt[1 - e^2*x^2]))])/((c + a*e^2)*Sqrt[2*c + a*e^2 + 
 2*Sqrt[c]*Sqrt[c + a*e^2]]))/(2*a^(3/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {643, 296, 291, 218}

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.

\(\displaystyle \int \frac {1}{\sqrt {1-e x} \sqrt {e x+1} \left (a+c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 643

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (2 a e^2+c\right ) \arctan \left (\frac {x \sqrt {a e^2+c}}{\sqrt {a} \sqrt {1-e^2 x^2}}\right )}{2 a^{3/2} \left (a e^2+c\right )^{3/2}}+\frac {c x \sqrt {1-e^2 x^2}}{2 a \left (a e^2+c\right ) \left (a+c x^2\right )}\)

Input:

Int[1/(Sqrt[1 - e*x]*Sqrt[1 + e*x]*(a + c*x^2)^2),x]
 

Output:

(c*x*Sqrt[1 - e^2*x^2])/(2*a*(c + a*e^2)*(a + c*x^2)) + ((c + 2*a*e^2)*Arc 
Tan[(Sqrt[c + a*e^2]*x)/(Sqrt[a]*Sqrt[1 - e^2*x^2])])/(2*a^(3/2)*(c + a*e^ 
2)^(3/2))
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 643

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_.), x_Symbol] :> Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x] /; FreeQ[{a 
, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && (Integer 
Q[m] || (GtQ[c, 0] && GtQ[e, 0]))
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.94 (sec) , antiderivative size = 973, normalized size of antiderivative = 9.45

method	result	size
default	\(-\frac {\left (2 \ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) a^{2} c \,e^{4} x^{2}-2 \ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) a^{2} c \,e^{4} x^{2}+2 \ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) a^{3} e^{4}+3 \ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) a \,c^{2} e^{2} x^{2}-2 \ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) a^{3} e^{4}-3 \ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) a \,c^{2} e^{2} x^{2}-2 a c \,e^{2} x \sqrt {-a c}\, \sqrt {-e^{2} x^{2}+1}\, \sqrt {\frac {a \,e^{2}+c}{c}}+3 \ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) a^{2} c \,e^{2}+\ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) c^{3} x^{2}-3 \ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) a^{2} c \,e^{2}-\ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) c^{3} x^{2}-2 c^{2} x \sqrt {-a c}\, \sqrt {-e^{2} x^{2}+1}\, \sqrt {\frac {a \,e^{2}+c}{c}}+\ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x -\sqrt {-a c}}\right ) a \,c^{2}-\ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {\frac {a \,e^{2}+c}{c}}\, \sqrt {-e^{2} x^{2}+1}\, c +2 c}{c x +\sqrt {-a c}}\right ) a \,c^{2}\right ) \operatorname {csgn}\left (e \right )^{2} c^{3} \sqrt {e x +1}\, \sqrt {-e x +1}}{4 \sqrt {-a c}\, \left (c x +\sqrt {-a c}\right ) \sqrt {\frac {a \,e^{2}+c}{c}}\, \left (c x -\sqrt {-a c}\right ) a \left (-e \sqrt {-a c}+c \right )^{2} \left (e \sqrt {-a c}+c \right )^{2} \sqrt {-e^{2} x^{2}+1}}\)	\(973\)

Input:

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

Output:

-1/4*(2*ln(2*(-(-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c 
+c)/(c*x-(-a*c)^(1/2)))*a^2*c*e^4*x^2-2*ln(2*((-a*c)^(1/2)*e^2*x+((a*e^2+c 
)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x+(-a*c)^(1/2)))*a^2*c*e^4*x^2+2*ln( 
2*(-(-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x-(- 
a*c)^(1/2)))*a^3*e^4+3*ln(2*(-(-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2 
*x^2+1)^(1/2)*c+c)/(c*x-(-a*c)^(1/2)))*a*c^2*e^2*x^2-2*ln(2*((-a*c)^(1/2)* 
e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x+(-a*c)^(1/2)))*a^3* 
e^4-3*ln(2*((-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c) 
/(c*x+(-a*c)^(1/2)))*a*c^2*e^2*x^2-2*a*c*e^2*x*(-a*c)^(1/2)*(-e^2*x^2+1)^( 
1/2)*((a*e^2+c)/c)^(1/2)+3*ln(2*(-(-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*( 
-e^2*x^2+1)^(1/2)*c+c)/(c*x-(-a*c)^(1/2)))*a^2*c*e^2+ln(2*(-(-a*c)^(1/2)*e 
^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x-(-a*c)^(1/2)))*c^3*x 
^2-3*ln(2*((-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/ 
(c*x+(-a*c)^(1/2)))*a^2*c*e^2-ln(2*((-a*c)^(1/2)*e^2*x+((a*e^2+c)/c)^(1/2) 
*(-e^2*x^2+1)^(1/2)*c+c)/(c*x+(-a*c)^(1/2)))*c^3*x^2-2*c^2*x*(-a*c)^(1/2)* 
(-e^2*x^2+1)^(1/2)*((a*e^2+c)/c)^(1/2)+ln(2*(-(-a*c)^(1/2)*e^2*x+((a*e^2+c 
)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x-(-a*c)^(1/2)))*a*c^2-ln(2*((-a*c)^ 
(1/2)*e^2*x+((a*e^2+c)/c)^(1/2)*(-e^2*x^2+1)^(1/2)*c+c)/(c*x+(-a*c)^(1/2)) 
)*a*c^2)*csgn(e)^2*c^3*(e*x+1)^(1/2)*(-e*x+1)^(1/2)/(-a*c)^(1/2)/(c*x+(-a* 
c)^(1/2))/((a*e^2+c)/c)^(1/2)/(c*x-(-a*c)^(1/2))/a/(-e*(-a*c)^(1/2)+c)^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (87) = 174\).

Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 3.65 \[ \int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\left [\frac {2 \, {\left (a^{2} c e^{2} + a c^{2}\right )} \sqrt {e x + 1} \sqrt {-e x + 1} x - {\left (2 \, a^{2} e^{2} + {\left (2 \, a c e^{2} + c^{2}\right )} x^{2} + a c\right )} \sqrt {-a^{2} e^{2} - a c} \log \left (-\frac {{\left (2 \, a e^{2} + c\right )} x^{2} - 2 \, \sqrt {-a^{2} e^{2} - a c} \sqrt {e x + 1} \sqrt {-e x + 1} x - a}{c x^{2} + a}\right )}{4 \, {\left (a^{5} e^{4} + 2 \, a^{4} c e^{2} + a^{3} c^{2} + {\left (a^{4} c e^{4} + 2 \, a^{3} c^{2} e^{2} + a^{2} c^{3}\right )} x^{2}\right )}}, \frac {{\left (a^{2} c e^{2} + a c^{2}\right )} \sqrt {e x + 1} \sqrt {-e x + 1} x - {\left (2 \, a^{2} e^{2} + {\left (2 \, a c e^{2} + c^{2}\right )} x^{2} + a c\right )} \sqrt {a^{2} e^{2} + a c} \arctan \left (\frac {\sqrt {a^{2} e^{2} + a c} \sqrt {e x + 1} \sqrt {-e x + 1} x}{a e^{2} x^{2} - a}\right )}{2 \, {\left (a^{5} e^{4} + 2 \, a^{4} c e^{2} + a^{3} c^{2} + {\left (a^{4} c e^{4} + 2 \, a^{3} c^{2} e^{2} + a^{2} c^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/4*(2*(a^2*c*e^2 + a*c^2)*sqrt(e*x + 1)*sqrt(-e*x + 1)*x - (2*a^2*e^2 + 
(2*a*c*e^2 + c^2)*x^2 + a*c)*sqrt(-a^2*e^2 - a*c)*log(-((2*a*e^2 + c)*x^2 
- 2*sqrt(-a^2*e^2 - a*c)*sqrt(e*x + 1)*sqrt(-e*x + 1)*x - a)/(c*x^2 + a))) 
/(a^5*e^4 + 2*a^4*c*e^2 + a^3*c^2 + (a^4*c*e^4 + 2*a^3*c^2*e^2 + a^2*c^3)* 
x^2), 1/2*((a^2*c*e^2 + a*c^2)*sqrt(e*x + 1)*sqrt(-e*x + 1)*x - (2*a^2*e^2 
 + (2*a*c*e^2 + c^2)*x^2 + a*c)*sqrt(a^2*e^2 + a*c)*arctan(sqrt(a^2*e^2 + 
a*c)*sqrt(e*x + 1)*sqrt(-e*x + 1)*x/(a*e^2*x^2 - a)))/(a^5*e^4 + 2*a^4*c*e 
^2 + a^3*c^2 + (a^4*c*e^4 + 2*a^3*c^2*e^2 + a^2*c^3)*x^2)]
 

Sympy [F]

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

integrate(1/(-e*x+1)**(1/2)/(e*x+1)**(1/2)/(c*x**2+a)**2,x)
 

Output:

Integral(1/((a + c*x**2)**2*sqrt(-e*x + 1)*sqrt(e*x + 1)), x)
 

Maxima [F]

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

integrate(1/(-e*x+1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a)^2,x, algorithm="maxima" 
)
 

Output:

integrate(1/((c*x^2 + a)^2*sqrt(e*x + 1)*sqrt(-e*x + 1)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2553 vs. \(2 (87) = 174\).

Time = 5.87 (sec) , antiderivative size = 2553, normalized size of antiderivative = 24.79 \[ \int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*e^3*((2*(2*a^3*c*e^4 + 2*a^2*c^2*e^4 + 3*a^2*c^2*e^2 + 3*a*c^3*e^2 + 
a*c^3 + c^4)*(a^2*e^5 + a*c*e^3)^2*sqrt(a^2*e^2 + a*c)*sgn(a^3*e^6 + 2*a^2 
*c*e^4 + a*c^2*e^2) - 2*(2*a^4*e^5 + 2*a^3*c*e^5 + 3*a^3*c*e^3 + 3*a^2*c^2 
*e^3 + a^2*c^2*e + a*c^3*e)*(a^2*e^5 + a*c*e^3)^2*sqrt(-a*c*e^2 - c^2) - ( 
2*a^6*e^10 + 2*a^5*c*e^10 + 7*a^5*c*e^8 + 7*a^4*c^2*e^8 + 9*a^4*c^2*e^6 + 
9*a^3*c^3*e^6 + 5*a^3*c^3*e^4 + 5*a^2*c^4*e^4 + a^2*c^4*e^2 + a*c^5*e^2)*s 
qrt(-a*c*e^2 - c^2)*abs(-a^2*e^5 - a*c*e^3)*sgn(a^3*e^6 + 2*a^2*c*e^4 + a* 
c^2*e^2) + (2*a^6*e^11 + 2*a^5*c*e^11 + 7*a^5*c*e^9 + 7*a^4*c^2*e^9 + 9*a^ 
4*c^2*e^7 + 9*a^3*c^3*e^7 + 5*a^3*c^3*e^5 + 5*a^2*c^4*e^5 + a^2*c^4*e^3 + 
a*c^5*e^3)*sqrt(a^2*e^2 + a*c)*abs(-a^2*e^5 - a*c*e^3) + (2*a^7*c*e^14 + 2 
*a^6*c^2*e^14 + 5*a^6*c^2*e^12 + 5*a^5*c^3*e^12 + 2*a^5*c^3*e^10 + 2*a^4*c 
^4*e^10 - 4*a^4*c^4*e^8 - 4*a^3*c^5*e^8 - 4*a^3*c^5*e^6 - 4*a^2*c^6*e^6 - 
a^2*c^6*e^4 - a*c^7*e^4)*sqrt(a^2*e^2 + a*c)*sgn(a^3*e^6 + 2*a^2*c*e^4 + a 
*c^2*e^2) - (2*a^8*e^15 + 2*a^7*c*e^15 + 5*a^7*c*e^13 + 5*a^6*c^2*e^13 + 2 
*a^6*c^2*e^11 + 2*a^5*c^3*e^11 - 4*a^5*c^3*e^9 - 4*a^4*c^4*e^9 - 4*a^4*c^4 
*e^7 - 4*a^3*c^5*e^7 - a^3*c^5*e^5 - a^2*c^6*e^5)*sqrt(-a*c*e^2 - c^2))*ar 
ctan(-1/2*((sqrt(2) - sqrt(-e*x + 1))/sqrt(e*x + 1) - sqrt(e*x + 1)/(sqrt( 
2) - sqrt(-e*x + 1)))/sqrt((a^3*e^6 - a*c^2*e^2 + sqrt(-(a^3*e^6 + 2*a^2*c 
*e^4 + a*c^2*e^2)^2 + (a^3*e^6 - a*c^2*e^2)^2))/(a^3*e^6 + 2*a^2*c*e^4 + a 
*c^2*e^2)))/(((a^9 + a^8*c)*e^16 + 6*(a^8*c + a^7*c^2)*e^14 + 15*(a^7*c...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\text {Hanged} \] Input:

int(1/((a + c*x^2)^2*(1 - e*x)^(1/2)*(e*x + 1)^(1/2)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

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

int(1/(-e*x+1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a)^2,x)
 

Output:

int(1/(sqrt(e*x + 1)*sqrt( - e*x + 1)*a**2 + 2*sqrt(e*x + 1)*sqrt( - e*x + 
 1)*a*c*x**2 + sqrt(e*x + 1)*sqrt( - e*x + 1)*c**2*x**4),x)