\(\int \frac {e^{n \text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^2} \, dx\) [1347]

Optimal result

Integrand size = 25, antiderivative size = 239 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}-\frac {a \left (6+4 n-n^2-n^3\right ) (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac {a \left (6+4 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n (2+n)}-\frac {2 a (1-a x)^{-n/2} (1+a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {1+a x}{1-a x}\right )}{c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.68 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (6 a x-6 a^3 x^3+n^3 (-1+a x)^2 (1+a x)+a n^2 x \left (-2+a^2 x^2\right )+n \left (-4+4 a x+6 a^2 x^2-4 a^3 x^3\right )-2 a n^2 (2+n) x (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (1,1-\frac {n}{2},2-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{c^2 (-2+n) n (2+n) x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {6700, 144, 25, 27, 172, 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.

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

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\int \frac {(1-a x)^{-\frac {n}{2}-2} (a x+1)^{\frac {n-4}{2}}}{x^2}dx}{c^2}\)

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

\(\displaystyle \frac {a \left (\frac {\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}-\frac {\int -\frac {a (1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} \left ((n+2) n^2+a \left (n^2+4 n+6\right ) x\right )}{x}dx}{a n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \left (\frac {\frac {\int \frac {a (1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} \left ((n+2) n^2+a \left (n^2+4 n+6\right ) x\right )}{x}dx}{a n}+\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {\frac {\int \frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-4}{2}} \left ((n+2) n^2+a \left (n^2+4 n+6\right ) x\right )}{x}dx}{n}+\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {a \left (\frac {\frac {\frac {\int \frac {a (2-n) n^2 (n+2) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{x}dx}{a (2-n)}-\frac {\left (-n^3-n^2+4 n+6\right ) (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{2-n}}{n}+\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {\frac {n^2 (n+2) \int \frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{x}dx-\frac {\left (-n^3-n^2+4 n+6\right ) (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{2-n}}{n}+\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {a \left (\frac {\frac {-2 n (n+2) (a x+1)^{n/2} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a x+1}{1-a x}\right )-\frac {\left (-n^3-n^2+4 n+6\right ) (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{2-n}}{n}+\frac {\left (n^2+4 n+6\right ) (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{n}}{n+2}+\frac {(n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{n+2}\right )-\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x}}{c^2}\)

Input:

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

Output:

(-(((1 - a*x)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/x) + a*(((3 + n)*(1 - a*x 
)^(-1 - n/2)*(1 + a*x)^((-2 + n)/2))/(2 + n) + (((6 + 4*n + n^2)*(1 + a*x) 
^((-2 + n)/2))/(n*(1 - a*x)^(n/2)) + (-(((6 + 4*n - n^2 - n^3)*(1 - a*x)^( 
1 - n/2)*(1 + a*x)^((-2 + n)/2))/(2 - n)) - (2*n*(2 + n)*(1 + a*x)^(n/2)*H 
ypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - a*x)^(n/2)) 
/n)/(2 + n)))/c^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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((-(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]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{6} - 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \] Input:

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

Output:

Integral(exp(n*atanh(a*x))/(a**4*x**6 - 2*a**2*x**4 + x**2), x)/c**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx=\frac {-6 e^{\mathit {atanh} \left (a x \right ) n} a^{3} x^{3}+6 e^{\mathit {atanh} \left (a x \right ) n} a^{2} n \,x^{2}-3 e^{\mathit {atanh} \left (a x \right ) n} a \,n^{2} x +6 e^{\mathit {atanh} \left (a x \right ) n} a x +e^{\mathit {atanh} \left (a x \right ) n} n^{3}-4 e^{\mathit {atanh} \left (a x \right ) n} n +\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a^{3} n^{4} x^{3}-4 \left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a^{3} n^{2} x^{3}-\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a \,n^{4} x +4 \left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \right ) a \,n^{2} x}{c^{2} n x \left (a^{2} n^{2} x^{2}-4 a^{2} x^{2}-n^{2}+4\right )} \] Input:

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

Output:

( - 6*e**(atanh(a*x)*n)*a**3*x**3 + 6*e**(atanh(a*x)*n)*a**2*n*x**2 - 3*e* 
*(atanh(a*x)*n)*a*n**2*x + 6*e**(atanh(a*x)*n)*a*x + e**(atanh(a*x)*n)*n** 
3 - 4*e**(atanh(a*x)*n)*n + int(e**(atanh(a*x)*n)/(a**4*x**5 - 2*a**2*x**3 
 + x),x)*a**3*n**4*x**3 - 4*int(e**(atanh(a*x)*n)/(a**4*x**5 - 2*a**2*x**3 
 + x),x)*a**3*n**2*x**3 - int(e**(atanh(a*x)*n)/(a**4*x**5 - 2*a**2*x**3 + 
 x),x)*a*n**4*x + 4*int(e**(atanh(a*x)*n)/(a**4*x**5 - 2*a**2*x**3 + x),x) 
*a*n**2*x)/(c**2*n*x*(a**2*n**2*x**2 - 4*a**2*x**2 - n**2 + 4))