\(\int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} (a+c x^2)^2} \, dx\) [18]

Optimal result

Integrand size = 28, antiderivative size = 137 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=-\frac {c x \sqrt {-1+e x} \sqrt {1+e x}}{2 a \left (c+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c+2 a e^2\right ) \sqrt {-1+e^2 x^2} \text {arctanh}\left (\frac {\sqrt {c+a e^2} x}{\sqrt {a} \sqrt {-1+e^2 x^2}}\right )}{2 a^{3/2} \left (c+a e^2\right )^{3/2} \sqrt {-1+e x} \sqrt {1+e x}} \] Output:

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

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=-\frac {c x \sqrt {-1+e x} \sqrt {1+e x}}{2 a \left (c+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (c+2 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c+a e^2} x}{\sqrt {a} \sqrt {-1+e x} \sqrt {1+e x}}\right )}{2 a^{3/2} \left (c+a e^2\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {648, 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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 296

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

Maple [C] (verified)

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

Time = 1.61 (sec) , antiderivative size = 996, normalized size of antiderivative = 7.27

method	result	size
default	\(\frac {\left (2 \ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, 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 {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x +\sqrt {-a c}}\right ) a \,c^{2}-\ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x -\sqrt {-a c}}\right ) a \,c^{2}\right ) c^{3} \operatorname {csgn}\left (e \right )^{2} \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}}\)	\(996\)

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (113) = 226\).

Time = 0.11 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.99 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\left [-\frac {2 \, a^{3} e^{3} + 2 \, a^{2} c e + 2 \, {\left (a^{2} c e^{2} + a c^{2}\right )} \sqrt {e x + 1} \sqrt {e x - 1} x + 2 \, {\left (a^{2} c e^{3} + a c^{2} e\right )} x^{2} - {\left (2 \, a^{2} e^{2} + {\left (2 \, a c e^{2} + c^{2}\right )} x^{2} + a c\right )} \sqrt {a^{2} e^{2} + a c} \log \left (-\frac {2 \, a^{2} e^{2} - {\left (4 \, a^{2} e^{4} + 4 \, a c e^{2} + c^{2}\right )} x^{2} - 2 \, {\left (\sqrt {a^{2} e^{2} + a c} {\left (2 \, a e^{2} + c\right )} x + 2 \, {\left (a^{2} e^{3} + a c e\right )} x\right )} \sqrt {e x + 1} \sqrt {e x - 1} + a c - 2 \, \sqrt {a^{2} e^{2} + a c} {\left ({\left (2 \, a e^{3} + c e\right )} x^{2} - a e\right )}}{c x^{2} + a}\right )}{4 \, {\left (a^{5} e^{4} + 2 \, a^{4} c e^{2} + a^{3} c^{2} + {\left (a^{4} c e^{4} + 2 \, a^{3} c^{2} e^{2} + a^{2} c^{3}\right )} x^{2}\right )}}, -\frac {a^{3} e^{3} + a^{2} c e + {\left (a^{2} c e^{2} + a c^{2}\right )} \sqrt {e x + 1} \sqrt {e x - 1} x + {\left (a^{2} c e^{3} + a c^{2} e\right )} x^{2} - {\left (2 \, a^{2} e^{2} + {\left (2 \, a c e^{2} + c^{2}\right )} x^{2} + a c\right )} \sqrt {-a^{2} e^{2} - a c} \arctan \left (\frac {\sqrt {-a^{2} e^{2} - a c} \sqrt {e x + 1} \sqrt {e x - 1} c x - \sqrt {-a^{2} e^{2} - a c} {\left (c e x^{2} + a e\right )}}{a^{2} e^{2} + a c}\right )}{2 \, {\left (a^{5} e^{4} + 2 \, a^{4} c e^{2} + a^{3} c^{2} + {\left (a^{4} c e^{4} + 2 \, a^{3} c^{2} e^{2} + a^{2} c^{3}\right )} x^{2}\right )}}\right ] \] 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^3*e^3 + 2*a^2*c*e + 2*(a^2*c*e^2 + a*c^2)*sqrt(e*x + 1)*sqrt(e* 
x - 1)*x + 2*(a^2*c*e^3 + a*c^2*e)*x^2 - (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^2*e^2 - (4*a^2*e^4 + 4*a*c*e^2 + c^ 
2)*x^2 - 2*(sqrt(a^2*e^2 + a*c)*(2*a*e^2 + c)*x + 2*(a^2*e^3 + a*c*e)*x)*s 
qrt(e*x + 1)*sqrt(e*x - 1) + a*c - 2*sqrt(a^2*e^2 + a*c)*((2*a*e^3 + c*e)* 
x^2 - a*e))/(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^3*e^3 + a^2*c*e + (a^2*c*e^2 + a*c^ 
2)*sqrt(e*x + 1)*sqrt(e*x - 1)*x + (a^2*c*e^3 + a*c^2*e)*x^2 - (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)*c*x - sqrt(-a^2*e^2 - a*c)*(c*e*x^2 + a 
*e))/(a^2*e^2 + a*c)))/(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. 243 vs. \(2 (113) = 226\).

Time = 0.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=-\frac {1}{2} \, e^{3} {\left (\frac {{\left (2 \, a e^{2} + c\right )} \arctan \left (\frac {c {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + 8 \, a e^{2} + 4 \, c}{8 \, \sqrt {-a^{2} e^{2} - a c} e}\right )}{{\left (a^{2} e^{4} + a c e^{2}\right )} \sqrt {-a^{2} e^{2} - a c} e} + \frac {8 \, {\left (2 \, a e^{2} {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + c {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + 4 \, c\right )}}{{\left (c {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{8} + 16 \, a e^{2} {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + 8 \, c {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + 16 \, c\right )} {\left (a^{2} e^{4} + a c e^{2}\right )}}\right )} \] 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*a*e^2 + c)*arctan(1/8*(c*(sqrt(e*x + 1) - sqrt(e*x - 1))^4 + 
8*a*e^2 + 4*c)/(sqrt(-a^2*e^2 - a*c)*e))/((a^2*e^4 + a*c*e^2)*sqrt(-a^2*e^ 
2 - a*c)*e) + 8*(2*a*e^2*(sqrt(e*x + 1) - sqrt(e*x - 1))^4 + c*(sqrt(e*x + 
 1) - sqrt(e*x - 1))^4 + 4*c)/((c*(sqrt(e*x + 1) - sqrt(e*x - 1))^8 + 16*a 
*e^2*(sqrt(e*x + 1) - sqrt(e*x - 1))^4 + 8*c*(sqrt(e*x + 1) - sqrt(e*x - 1 
))^4 + 16*c)*(a^2*e^4 + a*c*e^2)))
 

Mupad [B] (verification not implemented)

Time = 39.70 (sec) , antiderivative size = 14522, normalized size of antiderivative = 106.00 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-((c^2*atan(((a^(1/2)*c^5*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i)*32i)/(( 
e*x + 1)^(1/2) - 1) - (a^(1/2)*c^2*(c + a*e^2)^(9/2)*((e*x - 1)^(1/2) - 1i 
)*32i)/((e*x + 1)^(1/2) - 1) + (a^(1/2)*c^5*(c + a*e^2)^(3/2)*((e*x - 1)^( 
1/2) - 1i)^3*416i)/((e*x + 1)^(1/2) - 1)^3 - (a^(1/2)*c^2*(c + a*e^2)^(9/2 
)*((e*x - 1)^(1/2) - 1i)^3*416i)/((e*x + 1)^(1/2) - 1)^3 - (a^(5/2)*e^4*(c 
 + a*e^2)^(9/2)*((e*x - 1)^(1/2) - 1i)^3*40i)/((e*x + 1)^(1/2) - 1)^3 + (a 
^(11/2)*e^10*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i)^3*40i)/((e*x + 1)^(1 
/2) - 1)^3 - (a^(3/2)*c*e^2*(c + a*e^2)^(9/2)*((e*x - 1)^(1/2) - 1i)*32i)/ 
((e*x + 1)^(1/2) - 1) + (a^(9/2)*c*e^8*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) 
- 1i)*36i)/((e*x + 1)^(1/2) - 1) + (a^(3/2)*c^4*e^2*(c + a*e^2)^(3/2)*((e* 
x - 1)^(1/2) - 1i)*132i)/((e*x + 1)^(1/2) - 1) + (a^(5/2)*c^3*e^4*(c + a*e 
^2)^(3/2)*((e*x - 1)^(1/2) - 1i)*204i)/((e*x + 1)^(1/2) - 1) + (a^(7/2)*c^ 
2*e^6*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i)*140i)/((e*x + 1)^(1/2) - 1) 
 - (a^(3/2)*c*e^2*(c + a*e^2)^(9/2)*((e*x - 1)^(1/2) - 1i)^3*456i)/((e*x + 
 1)^(1/2) - 1)^3 + (a^(9/2)*c*e^8*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i) 
^3*580i)/((e*x + 1)^(1/2) - 1)^3 + (a^(3/2)*c^4*e^2*(c + a*e^2)^(3/2)*((e* 
x - 1)^(1/2) - 1i)^3*1708i)/((e*x + 1)^(1/2) - 1)^3 + (a^(5/2)*c^3*e^4*(c 
+ a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i)^3*2668i)/((e*x + 1)^(1/2) - 1)^3 + ( 
a^(7/2)*c^2*e^6*(c + a*e^2)^(3/2)*((e*x - 1)^(1/2) - 1i)^3*1916i)/((e*x + 
1)^(1/2) - 1)^3)/(a^6*c*e^11 + a^2*c^5*e^3 + 4*a^3*c^4*e^5 + 6*a^4*c^3*...
 

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)