\(\int \frac {e^{2 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{3/2}} \, dx\) [725]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {2}{a c \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {7}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{c^2}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a c^{3/2}} \] Output:

2/a/c/(c-c/a^2/x^2)^(1/2)+2/3*(a+1/x)/a^2/(c-c/a^2/x^2)^(3/2)+7/3/a^2/c/(c 
-c/a^2/x^2)^(1/2)/x-(c-c/a^2/x^2)^(1/2)*x/c^2-2*arctanh((c-c/a^2/x^2)^(1/2 
)/c^(1/2))/a/c^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {-10+4 a x+11 a^2 x^2-3 a^3 x^3-6 (-1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)} \] Input:

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2),x]
 

Output:

(-10 + 4*a*x + 11*a^2*x^2 - 3*a^3*x^3 - 6*(-1 + a*x)*Sqrt[-1 + a^2*x^2]*Lo 
g[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6709, 529, 2166, 27, 455, 223}

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.

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

\(\Big \downarrow \) 6709

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\int \frac {3 (a x+2)}{a^3 \sqrt {1-a^2 x^2}}dx-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\frac {3 \int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\frac {3 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{3} \left (\frac {3 \left (\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\)

Input:

Int[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2),x]
 

Output:

((1 - a^2*x^2)^(3/2)*((1 + a*x)^2/(3*a^4*(1 - a^2*x^2)^(3/2)) + ((-8*(1 + 
a*x))/(a^4*Sqrt[1 - a^2*x^2]) + (3*(-(Sqrt[1 - a^2*x^2]/a) + (2*ArcSin[a*x 
])/a))/a^3)/3))/((c - c/(a^2*x^2))^(3/2)*x^3)
 

Defintions of rubi rules used

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 6709

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p)   Int[u*((1 + a*x)^n/(x^ 
(2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[c, 0]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.64

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a^{2} c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}-\frac {\left (-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {8 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )}+\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {a^{2} c}}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\)	\(216\)
default	\(-\frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} x^{3}-15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} x^{2}+6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x -4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a x -6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a c +12 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}-2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}\right ) \left (a x +1\right )}{3 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\)	\(326\)

Input:

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

Output:

-1/a^2*(a^2*x^2-1)/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)-(-1/3/a^6/c/(x-1/a)^2 
*((x-1/a)^2*a^2*c+2*(x-1/a)*a*c)^(1/2)-8/3/a^5/c/(x-1/a)*((x-1/a)^2*a^2*c+ 
2*(x-1/a)*a*c)^(1/2)+2/a^3*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/( 
a^2*c)^(1/2))*a^2/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.13 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \] Input:

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

Output:

[1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2* 
sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*s 
qrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3*( 
6*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - 
 c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt(( 
a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-3 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+14 \sqrt {a^{2} x^{2}-1}\, a x -10 \sqrt {a^{2} x^{2}-1}-6 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{2} x^{2}+12 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x -6 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )-3 a^{2} x^{2}+6 a x -3\right )}{3 a \,c^{2} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:

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

Output:

(sqrt(c)*( - 3*sqrt(a**2*x**2 - 1)*a**2*x**2 + 14*sqrt(a**2*x**2 - 1)*a*x 
- 10*sqrt(a**2*x**2 - 1) - 6*log(sqrt(a**2*x**2 - 1) + a*x)*a**2*x**2 + 12 
*log(sqrt(a**2*x**2 - 1) + a*x)*a*x - 6*log(sqrt(a**2*x**2 - 1) + a*x) - 3 
*a**2*x**2 + 6*a*x - 3))/(3*a*c**2*(a**2*x**2 - 2*a*x + 1))