\(\int \frac {e^{n \text {arctanh}(a x)} x}{(c-a^2 c x^2)^2} \, dx\) [1344]

Optimal result

Integrand size = 23, antiderivative size = 69 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {e^{n \text {arctanh}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \text {arctanh}(a x)} (2-a n x)}{a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \] Output:

-exp(n*arctanh(a*x))/a^2/c^2/(-n^2+4)+exp(n*arctanh(a*x))*(-a*n*x+2)/a^2/c 
^2/(-n^2+4)/(-a^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (1-a n x+a^2 x^2\right )}{a^2 c^2 \left (-4+n^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.57, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6695, 27, 6686, 6687}

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.

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

\(\Big \downarrow \) 6695

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6686

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

\(\Big \downarrow \) 6687

\(\displaystyle \frac {e^{n \text {arctanh}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \left (\frac {2 e^{n \text {arctanh}(a x)}}{a n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{2 a c^2}\)

Input:

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

Output:

E^(n*ArcTanh[a*x])/(2*a^2*c^2*(1 - a^2*x^2)) - (n*((2*E^(n*ArcTanh[a*x]))/ 
(a*n*(4 - n^2)) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*(4 - n^2)*(1 - a^2*x 
^2))))/(2*a*c^2)
 

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 6686

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTanh[a*x])/(a*c*(n^2 - 
 4*(p + 1)^2))), x] - Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2))) 
Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x 
] && EqQ[a^2*c + d, 0] && LtQ[p, -1] &&  !IntegerQ[n] && NeQ[n^2 - 4*(p + 1 
)^2, 0] && IntegerQ[2*p]
 

rule 6687

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[ 
E^(n*ArcTanh[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 
0] &&  !IntegerQ[n/2]
 

rule 6695

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] 
 :> Simp[(c + d*x^2)^(p + 1)*(E^(n*ArcTanh[a*x])/(2*d*(p + 1))), x] - Simp[ 
a*c*(n/(2*d*(p + 1)))   Int[(c + d*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; Fre 
eQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] &&  !IntegerQ[n] && 
IntegerQ[2*p]
 

Maple [A] (verified)

Time = 6.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68

method	result	size
gosper	\(\frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (a^{2} x^{2}-n a x +1\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\)	\(47\)
orering	\(\frac {\left (a^{2} x^{2}-n a x +1\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a^{2} \left (n^{2}-4\right ) \left (-a^{2} c \,x^{2}+c \right )^{2}}\)	\(56\)
risch	\(\frac {\left (a^{2} x^{2}-n a x +1\right ) \left (a x +1\right )^{\frac {n}{2}} \left (-a x +1\right )^{-\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\)	\(59\)
parallelrisch	\(\frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{2} a^{2}-x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n +{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{c^{2} \left (a^{2} x^{2}-1\right ) a^{2} \left (n^{2}-4\right )}\)	\(60\)

Input:

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

Output:

exp(n*arctanh(a*x))*(a^2*x^2-a*n*x+1)/(a^2*x^2-1)/c^2/a^2/(n^2-4)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {{\left (a^{2} x^{2} - a n x + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right )} x^{2}} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {x^{2}}{2 c^{2}} & \text {for}\: a = 0 \\- \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} + \frac {a x}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\\frac {\int \frac {x e^{2 \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\\frac {a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} - \frac {a n x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} + \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**2/(2*c**2), Eq(a, 0)), (-a**2*x**2*atanh(a*x)/(4*a**4*c**2*x 
**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/ 
(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) + a*x 
/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - at 
anh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x 
))), Eq(n, -2)), (Integral(x*exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 
1), x)/c**2, Eq(n, 2)), (a**2*x**2*exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 
- 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) - a*n*x*exp(n*atanh(a*x 
))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) 
 + exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n 
**2 + 4*a**2*c**2), True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 26.72 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (a^2\,x^2-n\,a\,x+1\right )}{a^2\,c^2\,\left (n^2-4\right )\,\left (a^2\,x^2-1\right )} \] Input:

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

Output:

(exp(n*atanh(a*x))*(a^2*x^2 - a*n*x + 1))/(a^2*c^2*(n^2 - 4)*(a^2*x^2 - 1) 
)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} \left (a^{2} x^{2}-a n x +1\right )}{a^{2} c^{2} \left (a^{2} n^{2} x^{2}-4 a^{2} x^{2}-n^{2}+4\right )} \] Input:

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

Output:

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