\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-a^2 c x^2)^3} \, dx\) [1348]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {24 e^{n \text {arctanh}(a x)}}{a c^3 n \left (64-20 n^2+n^4\right )}-\frac {e^{n \text {arctanh}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \text {arctanh}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )} \] Output:

24*exp(n*arctanh(a*x))/a/c^3/n/(n^4-20*n^2+64)-exp(n*arctanh(a*x))*(-4*a*x 
+n)/a/c^3/(-n^2+16)/(-a^2*x^2+1)^2-12*exp(n*arctanh(a*x))*(-2*a*x+n)/a/c^3 
/(-n^2+4)/(-n^2+16)/(-a^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {(1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \left (n^4-4 a n^3 x+24 \left (-1+a^2 x^2\right )^2-8 a n x \left (-5+3 a^2 x^2\right )+4 n^2 \left (-4+3 a^2 x^2\right )\right )}{a c^3 (-4+n) (-2+n) n (2+n) (4+n)} \] Input:

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

Output:

((1 - a*x)^(-2 - n/2)*(1 + a*x)^(-2 + n/2)*(n^4 - 4*a*n^3*x + 24*(-1 + a^2 
*x^2)^2 - 8*a*n*x*(-5 + 3*a^2*x^2) + 4*n^2*(-4 + 3*a^2*x^2)))/(a*c^3*(-4 + 
 n)*(-2 + n)*n*(2 + n)*(4 + n))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6686, 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.

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

\(\Big \downarrow \) 6686

\(\displaystyle \frac {12 \int \frac {e^{n \text {arctanh}(a x)}}{c^2 \left (1-a^2 x^2\right )^2}dx}{c \left (16-n^2\right )}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {12 \int \frac {e^{n \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^2}dx}{c^3 \left (16-n^2\right )}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {12 \left (\frac {2 \int \frac {e^{n \text {arctanh}(a x)}}{1-a^2 x^2}dx}{4-n^2}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{c^3 \left (16-n^2\right )}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6687

\(\displaystyle \frac {12 \left (\frac {2 e^{n \text {arctanh}(a x)}}{a n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{c^3 \left (16-n^2\right )}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\)

Input:

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

Output:

-((E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2)) + ( 
12*((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))))/(c^3*(16 - n^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]
 

Maple [A] (verified)

Time = 25.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 n a x -16 n^{2}+24\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{2} c^{3} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n}\)	\(101\)
orering	\(-\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 n a x -16 n^{2}+24\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (n^{2}-16\right ) n a \left (n^{2}-4\right ) \left (-a^{2} c \,x^{2}+c \right )^{3}}\)	\(111\)
risch	\(\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 n a x -16 n^{2}+24\right ) \left (a x +1\right )^{\frac {n}{2}} \left (-a x +1\right )^{-\frac {n}{2}}}{\left (a^{2} x^{2}-1\right )^{2} c^{3} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n}\)	\(113\)
parallelrisch	\(\frac {24 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}+12 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2} n^{2}+40 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n -24 a^{3} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{3} n -4 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a \,n^{3}-48 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{2} a^{2}-16 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{2}+24 a^{4} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{4}+{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{4}}{c^{3} \left (a^{2} x^{2}-1\right )^{2} \left (n^{2}-16\right ) n a \left (n^{2}-4\right )}\)	\(159\)

Input:

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

Output:

(24*a^4*x^4-24*a^3*n*x^3+12*a^2*n^2*x^2-4*a*n^3*x-48*a^2*x^2+n^4+40*a*n*x- 
16*n^2+24)*exp(n*arctanh(a*x))/(a^2*x^2-1)^2/c^3/a/(n^2-16)/(n^2-4)/n
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.38 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {{\left (24 \, a^{4} x^{4} - 24 \, a^{3} n x^{3} + n^{4} + 12 \, {\left (a^{2} n^{2} - 4 \, a^{2}\right )} x^{2} - 16 \, n^{2} - 4 \, {\left (a n^{3} - 10 \, a n\right )} x + 24\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{5} - 20 \, a c^{3} n^{3} + 64 \, a c^{3} n + {\left (a^{5} c^{3} n^{5} - 20 \, a^{5} c^{3} n^{3} + 64 \, a^{5} c^{3} n\right )} x^{4} - 2 \, {\left (a^{3} c^{3} n^{5} - 20 \, a^{3} c^{3} n^{3} + 64 \, a^{3} c^{3} n\right )} x^{2}} \] Input:

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

Output:

(24*a^4*x^4 - 24*a^3*n*x^3 + n^4 + 12*(a^2*n^2 - 4*a^2)*x^2 - 16*n^2 - 4*( 
a*n^3 - 10*a*n)*x + 24)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n^5 - 20*a*c 
^3*n^3 + 64*a*c^3*n + (a^5*c^3*n^5 - 20*a^5*c^3*n^3 + 64*a^5*c^3*n)*x^4 - 
2*(a^3*c^3*n^5 - 20*a^3*c^3*n^3 + 64*a^3*c^3*n)*x^2)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 26.95 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.41 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {24\,x^4}{a\,c^3\,n\,\left (n^4-20\,n^2+64\right )}-\frac {4\,x\,\left (n^2-10\right )}{a^4\,c^3\,\left (n^4-20\,n^2+64\right )}-\frac {24\,x^3}{a^2\,c^3\,\left (n^4-20\,n^2+64\right )}+\frac {n^4-16\,n^2+24}{a^5\,c^3\,n\,\left (n^4-20\,n^2+64\right )}+\frac {x^2\,\left (12\,n^2-48\right )}{a^3\,c^3\,n\,\left (n^4-20\,n^2+64\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {1}{a^4}+x^4-\frac {2\,x^2}{a^2}\right )} \] Input:

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

Output:

((a*x + 1)^(n/2)*((24*x^4)/(a*c^3*n*(n^4 - 20*n^2 + 64)) - (4*x*(n^2 - 10) 
)/(a^4*c^3*(n^4 - 20*n^2 + 64)) - (24*x^3)/(a^2*c^3*(n^4 - 20*n^2 + 64)) + 
 (n^4 - 16*n^2 + 24)/(a^5*c^3*n*(n^4 - 20*n^2 + 64)) + (x^2*(12*n^2 - 48)) 
/(a^3*c^3*n*(n^4 - 20*n^2 + 64))))/((1 - a*x)^(n/2)*(1/a^4 + x^4 - (2*x^2) 
/a^2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.16 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} \left (24 a^{4} x^{4}-24 a^{3} n \,x^{3}+12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 a n x -16 n^{2}+24\right )}{a \,c^{3} n \left (a^{4} n^{4} x^{4}-20 a^{4} n^{2} x^{4}+64 a^{4} x^{4}-2 a^{2} n^{4} x^{2}+40 a^{2} n^{2} x^{2}-128 a^{2} x^{2}+n^{4}-20 n^{2}+64\right )} \] Input:

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

Output:

(e**(atanh(a*x)*n)*(24*a**4*x**4 - 24*a**3*n*x**3 + 12*a**2*n**2*x**2 - 48 
*a**2*x**2 - 4*a*n**3*x + 40*a*n*x + n**4 - 16*n**2 + 24))/(a*c**3*n*(a**4 
*n**4*x**4 - 20*a**4*n**2*x**4 + 64*a**4*x**4 - 2*a**2*n**4*x**2 + 40*a**2 
*n**2*x**2 - 128*a**2*x**2 + n**4 - 20*n**2 + 64))