\(\int \frac {(a+c x^2)^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx\) [21]

Optimal result

Integrand size = 28, antiderivative size = 85 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=-\frac {\left (c+a e^2\right )^2 x}{e^4 \sqrt {-1+e x} \sqrt {1+e x}}+\frac {c^2 x \sqrt {-1+e x} \sqrt {1+e x}}{2 e^4}+\frac {c \left (3 c+4 a e^2\right ) \text {arccosh}(e x)}{2 e^5} \] Output:

-(a*e^2+c)^2*x/e^4/(e*x-1)^(1/2)/(e*x+1)^(1/2)+1/2*c^2*x*(e*x-1)^(1/2)*(e* 
x+1)^(1/2)/e^4+1/2*c*(4*a*e^2+3*c)*arccosh(e*x)/e^5
 

Mathematica [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {\frac {e x \left (-4 a c e^2-2 a^2 e^4+c^2 \left (-3+e^2 x^2\right )\right )}{\sqrt {-1+e x} \sqrt {1+e x}}+2 c \left (3 c+4 a e^2\right ) \text {arctanh}\left (\sqrt {\frac {-1+e x}{1+e x}}\right )}{2 e^5} \] Input:

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

Output:

((e*x*(-4*a*c*e^2 - 2*a^2*e^4 + c^2*(-3 + e^2*x^2)))/(Sqrt[-1 + e*x]*Sqrt[ 
1 + e*x]) + 2*c*(3*c + 4*a*e^2)*ArcTanh[Sqrt[(-1 + e*x)/(1 + e*x)]])/(2*e^ 
5)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {650, 2124, 25, 27, 646, 43}

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.

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

\(\Big \downarrow \) 650

\(\displaystyle -\frac {\int \frac {-c^2 x^3-\frac {c^2 x^2}{e}-c \left (2 a+\frac {c}{e^2}\right ) x+a^2 e}{\sqrt {e x-1} (e x+1)^{3/2}}dx}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {\frac {\int -\frac {c \left (c x^2+2 a+\frac {c}{e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}dx}{e}+\frac {\sqrt {e x-1} \left (a e^2+c\right )^2}{e^4 \sqrt {e x+1}}}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\sqrt {e x-1} \left (a e^2+c\right )^2}{e^4 \sqrt {e x+1}}-\frac {\int \frac {c \left (c x^2+2 a+\frac {c}{e^2}\right )}{\sqrt {e x-1} \sqrt {e x+1}}dx}{e}}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\sqrt {e x-1} \left (a e^2+c\right )^2}{e^4 \sqrt {e x+1}}-\frac {c \int \frac {c x^2+2 a+\frac {c}{e^2}}{\sqrt {e x-1} \sqrt {e x+1}}dx}{e}}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 646

\(\displaystyle -\frac {\frac {\sqrt {e x-1} \left (a e^2+c\right )^2}{e^4 \sqrt {e x+1}}-\frac {c \left (\frac {1}{2} \left (4 a+\frac {3 c}{e^2}\right ) \int \frac {1}{\sqrt {e x-1} \sqrt {e x+1}}dx+\frac {c x \sqrt {e x-1} \sqrt {e x+1}}{2 e^2}\right )}{e}}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 43

\(\displaystyle -\frac {\frac {\sqrt {e x-1} \left (a e^2+c\right )^2}{e^4 \sqrt {e x+1}}-\frac {c \left (\frac {\left (4 a+\frac {3 c}{e^2}\right ) \text {arccosh}(e x)}{2 e}+\frac {c x \sqrt {e x-1} \sqrt {e x+1}}{2 e^2}\right )}{e}}{e}-\frac {\left (a e^2+c\right )^2}{e^5 \sqrt {e x-1} \sqrt {e x+1}}\)

Input:

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

Output:

-((c + a*e^2)^2/(e^5*Sqrt[-1 + e*x]*Sqrt[1 + e*x])) - (((c + a*e^2)^2*Sqrt 
[-1 + e*x])/(e^4*Sqrt[1 + e*x]) - (c*((c*x*Sqrt[-1 + e*x]*Sqrt[1 + e*x])/( 
2*e^2) + ((4*a + (3*c)/e^2)*ArcCosh[e*x])/(2*e)))/e)/e
 

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 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 646

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2), x_Symbol] :> Simp[b*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 1)/(d*f*(2*m + 
 3))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(d*f*(2*m + 3))   Int[(c + d*x)^ 
m*(e + f*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m, n] && 
 EqQ[d*e + c*f, 0] &&  !LtQ[m, -1]
 

rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + c*x^2)^p, d + e*x 
, x], R = PolynomialRemainder[(a + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x) 
^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Simp[1/((m + 1)*(e 
*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d 
*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g, n}, x] && 
 IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(73)=146\).

Time = 0.77 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.80

method	result	size
risch	\(\frac {c^{2} x \sqrt {e x -1}\, \sqrt {e x +1}}{2 e^{4}}+\frac {\left (\frac {\left (-a^{2} e^{4}-2 a c \,e^{2}-c^{2}\right ) \sqrt {e^{2} \left (x +\frac {1}{e}\right )^{2}-2 e \left (x +\frac {1}{e}\right )}}{e^{2} \left (x +\frac {1}{e}\right )}-\frac {\left (a^{2} e^{4}+2 a c \,e^{2}+c^{2}\right ) \sqrt {e^{2} \left (x -\frac {1}{e}\right )^{2}+2 e \left (x -\frac {1}{e}\right )}}{e^{2} \left (x -\frac {1}{e}\right )}+\frac {3 c^{2} \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right )}{\sqrt {e^{2}}}+\frac {4 a c \,e^{2} \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right )}{\sqrt {e^{2}}}\right ) \sqrt {\left (e x -1\right ) \left (e x +1\right )}}{2 e^{4} \sqrt {e x -1}\, \sqrt {e x +1}}\)	\(238\)
default	\(-\frac {\left (2 \,\operatorname {csgn}\left (e \right ) e^{5} \sqrt {e^{2} x^{2}-1}\, a^{2} x -\operatorname {csgn}\left (e \right ) c^{2} e^{3} x^{3} \sqrt {e^{2} x^{2}-1}-4 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a c \,e^{4} x^{2}+4 \,\operatorname {csgn}\left (e \right ) e^{3} \sqrt {e^{2} x^{2}-1}\, a c x -3 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) c^{2} e^{2} x^{2}+3 \,\operatorname {csgn}\left (e \right ) e \sqrt {e^{2} x^{2}-1}\, c^{2} x +4 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a c \,e^{2}+3 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) c^{2}\right ) \operatorname {csgn}\left (e \right )}{2 \sqrt {e^{2} x^{2}-1}\, e^{5} \sqrt {e x +1}\, \sqrt {e x -1}}\)	\(242\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (73) = 146\).

Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {2 \, a^{2} e^{4} + 4 \, a c e^{2} - 2 \, {\left (a^{2} e^{6} + 2 \, a c e^{4} + c^{2} e^{2}\right )} x^{2} + {\left (c^{2} e^{3} x^{3} - {\left (2 \, a^{2} e^{5} + 4 \, a c e^{3} + 3 \, c^{2} e\right )} x\right )} \sqrt {e x + 1} \sqrt {e x - 1} + 2 \, c^{2} + {\left (4 \, a c e^{2} - {\left (4 \, a c e^{4} + 3 \, c^{2} e^{2}\right )} x^{2} + 3 \, c^{2}\right )} \log \left (-e x + \sqrt {e x + 1} \sqrt {e x - 1}\right )}{2 \, {\left (e^{7} x^{2} - e^{5}\right )}} \] Input:

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

Output:

1/2*(2*a^2*e^4 + 4*a*c*e^2 - 2*(a^2*e^6 + 2*a*c*e^4 + c^2*e^2)*x^2 + (c^2* 
e^3*x^3 - (2*a^2*e^5 + 4*a*c*e^3 + 3*c^2*e)*x)*sqrt(e*x + 1)*sqrt(e*x - 1) 
 + 2*c^2 + (4*a*c*e^2 - (4*a*c*e^4 + 3*c^2*e^2)*x^2 + 3*c^2)*log(-e*x + sq 
rt(e*x + 1)*sqrt(e*x - 1)))/(e^7*x^2 - e^5)
 

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{2}}{\left (e x - 1\right )^{\frac {3}{2}} \left (e x + 1\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((c*x**2+a)**2/(e*x-1)**(3/2)/(e*x+1)**(3/2),x)
 

Output:

Integral((a + c*x**2)**2/((e*x - 1)**(3/2)*(e*x + 1)**(3/2)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (73) = 146\).

Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=-\frac {a^{2} x}{\sqrt {e^{2} x^{2} - 1}} + \frac {c^{2} x^{3}}{2 \, \sqrt {e^{2} x^{2} - 1} e^{2}} - \frac {2 \, a c x}{\sqrt {e^{2} x^{2} - 1} e^{2}} + \frac {2 \, a c \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{\sqrt {e^{2}} e^{2}} - \frac {3 \, c^{2} x}{2 \, \sqrt {e^{2} x^{2} - 1} e^{4}} + \frac {3 \, c^{2} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{2 \, \sqrt {e^{2}} e^{4}} \] Input:

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

Output:

-a^2*x/sqrt(e^2*x^2 - 1) + 1/2*c^2*x^3/(sqrt(e^2*x^2 - 1)*e^2) - 2*a*c*x/( 
sqrt(e^2*x^2 - 1)*e^2) + 2*a*c*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2) 
)/(sqrt(e^2)*e^2) - 3/2*c^2*x/(sqrt(e^2*x^2 - 1)*e^4) + 3/2*c^2*log(2*e^2* 
x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2))/(sqrt(e^2)*e^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (73) = 146\).

Time = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {\sqrt {e x + 1} {\left ({\left (e x + 1\right )} {\left (\frac {{\left (e x + 1\right )} c^{2}}{e^{5}} - \frac {3 \, c^{2}}{e^{5}}\right )} - \frac {a^{2} e^{19} + 2 \, a c e^{17} - c^{2} e^{15}}{e^{20}}\right )}}{2 \, \sqrt {e x - 1}} - \frac {{\left (4 \, a c e^{2} + 3 \, c^{2}\right )} \log \left ({\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{2}\right )}{2 \, e^{5}} - \frac {2 \, {\left (a^{2} e^{4} + 2 \, a c e^{2} + c^{2}\right )}}{{\left ({\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{2} + 2\right )} e^{5}} \] Input:

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

Output:

1/2*sqrt(e*x + 1)*((e*x + 1)*((e*x + 1)*c^2/e^5 - 3*c^2/e^5) - (a^2*e^19 + 
 2*a*c*e^17 - c^2*e^15)/e^20)/sqrt(e*x - 1) - 1/2*(4*a*c*e^2 + 3*c^2)*log( 
(sqrt(e*x + 1) - sqrt(e*x - 1))^2)/e^5 - 2*(a^2*e^4 + 2*a*c*e^2 + c^2)/((( 
sqrt(e*x + 1) - sqrt(e*x - 1))^2 + 2)*e^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^2}{{\left (e\,x-1\right )}^{3/2}\,{\left (e\,x+1\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.38 \[ \int \frac {\left (a+c x^2\right )^2}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {32 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a c \,e^{3} x +32 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a c \,e^{2}+24 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) c^{2} e x +24 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) c^{2}-8 \sqrt {e x -1}\, a^{2} e^{5} x -8 \sqrt {e x -1}\, a^{2} e^{4}-16 \sqrt {e x -1}\, a c \,e^{3} x -16 \sqrt {e x -1}\, a c \,e^{2}-9 \sqrt {e x -1}\, c^{2} e x -9 \sqrt {e x -1}\, c^{2}-8 \sqrt {e x +1}\, a^{2} e^{5} x -16 \sqrt {e x +1}\, a c \,e^{3} x +4 \sqrt {e x +1}\, c^{2} e^{3} x^{3}-12 \sqrt {e x +1}\, c^{2} e x}{8 \sqrt {e x -1}\, e^{5} \left (e x +1\right )} \] Input:

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

Output:

(32*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a*c*e**3*x 
+ 32*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a*c*e**2 + 
 24*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*c**2*e*x + 
24*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*c**2 - 8*sqr 
t(e*x - 1)*a**2*e**5*x - 8*sqrt(e*x - 1)*a**2*e**4 - 16*sqrt(e*x - 1)*a*c* 
e**3*x - 16*sqrt(e*x - 1)*a*c*e**2 - 9*sqrt(e*x - 1)*c**2*e*x - 9*sqrt(e*x 
 - 1)*c**2 - 8*sqrt(e*x + 1)*a**2*e**5*x - 16*sqrt(e*x + 1)*a*c*e**3*x + 4 
*sqrt(e*x + 1)*c**2*e**3*x**3 - 12*sqrt(e*x + 1)*c**2*e*x)/(8*sqrt(e*x - 1 
)*e**5*(e*x + 1))