\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [1349]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [A] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [F]
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 22, antiderivative size = 197 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {720 e^{n \text {arctanh}(a x)}}{a c^4 n \left (36-n^2\right ) \left (64-20 n^2+n^4\right )}-\frac {e^{n \text {arctanh}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \text {arctanh}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \text {arctanh}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )} \] Output:

720*exp(n*arctanh(a*x))/a/c^4/n/(-n^2+36)/(n^4-20*n^2+64)-exp(n*arctanh(a* 
x))*(-6*a*x+n)/a/c^4/(-n^2+36)/(-a^2*x^2+1)^3-30*exp(n*arctanh(a*x))*(-4*a 
*x+n)/a/c^4/(-n^2+16)/(-n^2+36)/(-a^2*x^2+1)^2-360*exp(n*arctanh(a*x))*(-2 
*a*x+n)/a/c^4/(-n^2+4)/(-n^2+16)/(-n^2+36)/(-a^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.86 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {(1-a x)^{-3-\frac {n}{2}} (1+a x)^{-3+\frac {n}{2}} \left (-n^6+6 a n^5 x+n^4 \left (50-30 a^2 x^2\right )+120 a n^3 x \left (-2+a^2 x^2\right )-720 \left (-1+a^2 x^2\right )^3+48 a n x \left (33-40 a^2 x^2+15 a^4 x^4\right )-8 n^2 \left (68-105 a^2 x^2+45 a^4 x^4\right )\right )}{a c^4 (-6+n) (-4+n) (-2+n) n (2+n) (4+n) (6+n)} \] Input:

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

Output:

-(((1 - a*x)^(-3 - n/2)*(1 + a*x)^(-3 + n/2)*(-n^6 + 6*a*n^5*x + n^4*(50 - 
 30*a^2*x^2) + 120*a*n^3*x*(-2 + a^2*x^2) - 720*(-1 + a^2*x^2)^3 + 48*a*n* 
x*(33 - 40*a^2*x^2 + 15*a^4*x^4) - 8*n^2*(68 - 105*a^2*x^2 + 45*a^4*x^4))) 
/(a*c^4*(-6 + n)*(-4 + n)*(-2 + n)*n*(2 + n)*(4 + n)*(6 + n)))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6686, 27, 6686, 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.

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

\(\Big \downarrow \) 6686

\(\displaystyle \frac {30 \int \frac {e^{n \text {arctanh}(a x)}}{c^3 \left (1-a^2 x^2\right )^3}dx}{c \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \text {arctanh}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {30 \int \frac {e^{n \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^3}dx}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \text {arctanh}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {30 \left (\frac {12 \int \frac {e^{n \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^2}dx}{16-n^2}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \text {arctanh}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {30 \left (\frac {12 \left (\frac {2 \int \frac {e^{n \text {arctanh}(a x)}}{1-a^2 x^2}dx}{4-n^2}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{16-n^2}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \text {arctanh}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\)

\(\Big \downarrow \) 6687

\(\displaystyle \frac {30 \left (\frac {12 \left (\frac {2 e^{n \text {arctanh}(a x)}}{a n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{16-n^2}-\frac {(n-4 a x) e^{n \text {arctanh}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \text {arctanh}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\)

Input:

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

Output:

-((E^(n*ArcTanh[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3)) + ( 
30*(-((E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*(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))))/(16 - n^2)))/(c^4*(36 - 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 = 96.63 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85

method result size
gosper \(-\frac {\left (720 x^{6} a^{6}-720 x^{5} a^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 n a x +544 n^{2}-720\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{3} c^{4} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\) \(167\)
orering \(-\frac {\left (720 x^{6} a^{6}-720 x^{5} a^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 n a x +544 n^{2}-720\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (n^{4}-40 n^{2}+144\right ) \left (n^{2}-16\right ) a n \left (-a^{2} c \,x^{2}+c \right )^{4}}\) \(178\)
risch \(-\frac {\left (720 x^{6} a^{6}-720 x^{5} a^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 n a x +544 n^{2}-720\right ) \left (a x +1\right )^{\frac {n}{2}} \left (-a x +1\right )^{-\frac {n}{2}}}{c^{4} \left (n^{2}-36\right ) \left (n^{2}-16\right ) \left (n^{2}-4\right ) a n \left (a^{2} x^{2}-1\right )^{3}}\) \(183\)
parallelrisch \(\frac {720 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}+840 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2} n^{2}+1584 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n -360 x^{4} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{4} n^{2}+120 x^{3} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{3} n^{3}-30 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2} n^{4}+6 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a \,n^{5}+720 a^{5} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{5} n -{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{6}-720 a^{6} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{6}-1920 a^{3} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{3} n -240 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a \,n^{3}-2160 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{2} a^{2}-544 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{2}+2160 a^{4} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{4}+50 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{4}}{c^{4} \left (a^{2} x^{2}-1\right )^{3} \left (n^{4}-40 n^{2}+144\right ) \left (n^{2}-16\right ) a n}\) \(276\)

Input:

int(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-(720*a^6*x^6-720*a^5*n*x^5+360*a^4*n^2*x^4-120*a^3*n^3*x^3-2160*a^4*x^4+3 
0*a^2*n^4*x^2+1920*a^3*n*x^3-6*a*n^5*x-840*a^2*n^2*x^2+n^6+240*a*n^3*x+216 
0*a^2*x^2-50*n^4-1584*a*n*x+544*n^2-720)*exp(n*arctanh(a*x))/(a^2*x^2-1)^3 
/c^4/a/n/(n^6-56*n^4+784*n^2-2304)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.57 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (720 \, a^{6} x^{6} - 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} - 120 \, {\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} - 6 \, {\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} - {\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right )} x^{2}} \] Input:

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="fricas")
 

Output:

(720*a^6*x^6 - 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 - 6*a^4)*x^4 - 50*n^4 - 
120*(a^3*n^3 - 16*a^3*n)*x^3 + 30*(a^2*n^4 - 28*a^2*n^2 + 72*a^2)*x^2 + 54 
4*n^2 - 6*(a*n^5 - 40*a*n^3 + 264*a*n)*x - 720)*(-(a*x + 1)/(a*x - 1))^(1/ 
2*n)/(a*c^4*n^7 - 56*a*c^4*n^5 + 784*a*c^4*n^3 - (a^7*c^4*n^7 - 56*a^7*c^4 
*n^5 + 784*a^7*c^4*n^3 - 2304*a^7*c^4*n)*x^6 - 2304*a*c^4*n + 3*(a^5*c^4*n 
^7 - 56*a^5*c^4*n^5 + 784*a^5*c^4*n^3 - 2304*a^5*c^4*n)*x^4 - 3*(a^3*c^4*n 
^7 - 56*a^3*c^4*n^5 + 784*a^3*c^4*n^3 - 2304*a^3*c^4*n)*x^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^4,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 27.19 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.53 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {n^6-50\,n^4+544\,n^2-720}{a^7\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {720\,x^5}{a^2\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {x^3\,\left (120\,n^2-1920\right )}{a^4\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {6\,x\,\left (n^4-40\,n^2+264\right )}{a^6\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^2\,\left (30\,n^4-840\,n^2+2160\right )}{a^5\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^4\,\left (360\,n^2-2160\right )}{a^3\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {1}{a^6}-x^6+\frac {3\,x^4}{a^2}-\frac {3\,x^2}{a^4}\right )} \] Input:

int(exp(n*atanh(a*x))/(c - a^2*c*x^2)^4,x)
 

Output:

((a*x + 1)^(n/2)*((544*n^2 - 50*n^4 + n^6 - 720)/(a^7*c^4*n*(784*n^2 - 56* 
n^4 + n^6 - 2304)) - (720*x^5)/(a^2*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) - 
 (x^3*(120*n^2 - 1920))/(a^4*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) + (720*x 
^6)/(a*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) - (6*x*(n^4 - 40*n^2 + 264)) 
/(a^6*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) + (x^2*(30*n^4 - 840*n^2 + 2160 
))/(a^5*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) + (x^4*(360*n^2 - 2160))/(a 
^3*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304))))/((1 - a*x)^(n/2)*(1/a^6 - x^6 
+ (3*x^4)/a^2 - (3*x^2)/a^4))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.43 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} \left (-720 a^{6} x^{6}+720 a^{5} n \,x^{5}-360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}-30 a^{2} n^{4} x^{2}-1920 a^{3} n \,x^{3}+6 a \,n^{5} x +840 a^{2} n^{2} x^{2}-n^{6}-240 a \,n^{3} x -2160 a^{2} x^{2}+50 n^{4}+1584 a n x -544 n^{2}+720\right )}{a \,c^{4} n \left (a^{6} n^{6} x^{6}-56 a^{6} n^{4} x^{6}+784 a^{6} n^{2} x^{6}-3 a^{4} n^{6} x^{4}-2304 a^{6} x^{6}+168 a^{4} n^{4} x^{4}-2352 a^{4} n^{2} x^{4}+3 a^{2} n^{6} x^{2}+6912 a^{4} x^{4}-168 a^{2} n^{4} x^{2}+2352 a^{2} n^{2} x^{2}-n^{6}-6912 a^{2} x^{2}+56 n^{4}-784 n^{2}+2304\right )} \] Input:

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

Output:

(e**(atanh(a*x)*n)*( - 720*a**6*x**6 + 720*a**5*n*x**5 - 360*a**4*n**2*x** 
4 + 2160*a**4*x**4 + 120*a**3*n**3*x**3 - 1920*a**3*n*x**3 - 30*a**2*n**4* 
x**2 + 840*a**2*n**2*x**2 - 2160*a**2*x**2 + 6*a*n**5*x - 240*a*n**3*x + 1 
584*a*n*x - n**6 + 50*n**4 - 544*n**2 + 720))/(a*c**4*n*(a**6*n**6*x**6 - 
56*a**6*n**4*x**6 + 784*a**6*n**2*x**6 - 2304*a**6*x**6 - 3*a**4*n**6*x**4 
 + 168*a**4*n**4*x**4 - 2352*a**4*n**2*x**4 + 6912*a**4*x**4 + 3*a**2*n**6 
*x**2 - 168*a**2*n**4*x**2 + 2352*a**2*n**2*x**2 - 6912*a**2*x**2 - n**6 + 
 56*n**4 - 784*n**2 + 2304))