\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{3/2}} \, dx\) [784]

Optimal result

Integrand size = 24, antiderivative size = 322 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {(2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^4 (1+n) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \left (1-a^2 x^2\right )^{3/2}}{a^4 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2^{\frac {1+n}{2}} n (1-a x)^{\frac {1-n}{2}} \left (1-a^2 x^2\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {1}{2} (1-a x)\right )}{a^4 (1-n) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \] Output:

(2+n)*(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(3/2)/a^4/(1 
+n)/(c-c/a^2/x^2)^(3/2)/x^3-(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a 
^2*x^2+1)^(3/2)/a^2/(c-c/a^2/x^2)^(3/2)/x+n*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^( 
-1/2+1/2*n)*(-a^2*x^2+1)^(3/2)/a^4/(-n^2+1)/(c-c/a^2/x^2)^(3/2)/x^3+2^(1/2 
+1/2*n)*n*(-a*x+1)^(1/2-1/2*n)*(-a^2*x^2+1)^(3/2)*hypergeom([1/2-1/2*n, 1/ 
2-1/2*n],[3/2-1/2*n],-1/2*a*x+1/2)/a^4/(1-n)/(c-c/a^2/x^2)^(3/2)/x^3
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.58 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {(1-a x)^{-\frac {1}{2}-\frac {n}{2}} \left (1-a^2 x^2\right )^{3/2} \left (-4 a^4 x^2 (1+a x)^{\frac {1}{2} (-1+n)}+\frac {4 a^2 (1+a x)^{\frac {1}{2} (-1+n)} \left (-2+n^2 (-1+2 a x)+n (-2+3 a x)\right )}{-1+n^2}+\frac {2^{\frac {3+n}{2}} a^2 n (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )}{-3+n}\right )}{4 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \] Input:

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

Output:

((1 - a*x)^(-1/2 - n/2)*(1 - a^2*x^2)^(3/2)*(-4*a^4*x^2*(1 + a*x)^((-1 + n 
)/2) + (4*a^2*(1 + a*x)^((-1 + n)/2)*(-2 + n^2*(-1 + 2*a*x) + n*(-2 + 3*a* 
x)))/(-1 + n^2) + (2^((3 + n)/2)*a^2*n*(-1 + a*x)^2*Hypergeometric2F1[3/2 
- n/2, 3/2 - n/2, 5/2 - n/2, 1/2 - (a*x)/2])/(-3 + n)))/(4*a^6*(c - c/(a^2 
*x^2))^(3/2)*x^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.66, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6710, 6700, 111, 25, 162, 79}

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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}} \left (-a (2 n+3) n x+n^2+2 n+2\right )}{a^2 \left (1-n^2\right )}-\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {3-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {3-n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (3-n)}}{a^2}-\frac {x^2 (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{a^2}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\)

Input:

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

Output:

((1 - a^2*x^2)^(3/2)*(-((x^2*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2) 
)/a^2) + (((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*(2 + 2*n + n^2 - 
a*n*(3 + 2*n)*x))/(a^2*(1 - n^2)) - (2^((-1 + n)/2)*n*(1 - a*x)^((3 - n)/2 
)*Hypergeometric2F1[(3 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(a^2*(3 
 - n)))/a^2))/((c - c/(a^2*x^2))^(3/2)*x^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 111

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

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

rule 6700

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

rule 6710

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/x^(2*p))*(1 - a 
^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {3}{2}}}d x\]

Input:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(3/2),x)
 

Output:

int(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(3/2),x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(3/2),x, algorithm="fricas")
 

Output:

integral(a^4*x^4*(-(a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2* 
x^2))/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)
 

Sympy [F]

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

integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**(3/2),x)
 

Output:

Integral(exp(n*atanh(a*x))/(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2), x)
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))/(c-c/a^2/x^2)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate(exp(n*arctanh(a*x))/(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. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^(3/2),x)
 

Output:

int(exp(n*atanh(a*x))/(c - c/(a^2*x^2))^(3/2), x)
 

Reduce [F]

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

int(exp(n*atanh(a*x))/(c-c/a^2/x^2)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(int((e**(atanh(a*x)*n)*x**3)/(sqrt(a**2*x**2 - 1)*a**2*x**2 - sqrt(a**2*x 
**2 - 1)),x)*a**3)/(sqrt(c)*c)