\(\int \frac {e^{n \text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^{3/2}} \, dx\) [1384]

Optimal result

Integrand size = 27, antiderivative size = 321 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {a (2+n) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \left (2+2 n+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {1+a x}{1-a x}\right )}{c (1-n) \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.54 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (-\left ((-3+n) (1+a x) \left (-1+2 a^2 x^2+n^2 (-1+a x)^2+a n x (-3+2 a x)\right )\right )+2 a n \left (-1+n^2\right ) x (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{c (-3+n) (-1+n) (1+n) x \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.74, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6703, 6700, 144, 25, 27, 172, 25, 27, 172, 27, 141}

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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(-(((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/x) + 
 a*(((2 + n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(1 + n) + (-(( 
(2 + 2*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(1 - n)) + ( 
2*n*(1 - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Hypergeometric2 
F1[1, (-1 + n)/2, (1 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - n)^2)/(1 + n))))/( 
c*Sqrt[c - a^2*c*x^2])
 

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 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[b*(a + b*x)^(m + 1)*( 
c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] 
+ Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) 
- b*d*f*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f 
, m, n, p}, x] && NeQ[m, -1]
 

rule 172

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

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 6703

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^6 
- 2*a^2*c^2*x^4 + c^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{4}-\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x}{\sqrt {c}\, c} \] Input:

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

Output:

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