\(\int e^{n \text {arctanh}(a x)} x^2 (c-a^2 c x^2)^{-1-\frac {n^2}{2}} \, dx\) [1405]

Optimal result

Integrand size = 33, antiderivative size = 53 \[ \int e^{n \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {e^{n \text {arctanh}(a x)} (1-a n x) \left (c-a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \] Output:

exp(n*arctanh(a*x))*(-a*n*x+1)/a^3/c/n/(-n^2+1)/((-a^2*c*x^2+c)^(1/2*n^2))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.68 \[ \int e^{n \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {(1-a x)^{-\frac {1}{2} n (1+n)} (1+a x)^{-\frac {1}{2} (-1+n) n} (-1+a n x) \left (1-a^2 x^2\right )^{\frac {n^2}{2}} \left (c-a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (-1+n^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6696}

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.

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

\(\Big \downarrow \) 6696

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 6696

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

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (n a x -1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\)	\(58\)
orering	\(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (n a x -1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\)	\(58\)
parallelrisch	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{3} a^{3} n -{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{2} \left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} a^{2}-\left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n +\left (-a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a^{3} n \left (n^{2}-1\right )}\)	\(143\)
risch	\(\frac {\left (a^{3} x^{3} n -a^{2} x^{2}-n a x +1\right ) {\mathrm e}^{\frac {n \ln \left (a x +1\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i \left (a x +1\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} n^{2}}{4}-\frac {i \pi \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i \left (a x +1\right )\right ) n^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} n^{2}}{4}-\frac {i \pi \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i c \right ) n^{2}}{4}-\frac {\ln \left (a x -1\right ) n}{2}-\frac {\ln \left (a x +1\right ) n^{2}}{2}-\frac {\ln \left (a x -1\right ) n^{2}}{2}-\ln \left (c \right )-\ln \left (a x +1\right )-\ln \left (a x -1\right )-\frac {n^{2} \ln \left (c \right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \right ) n^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i \left (a x +1\right )\right ) n^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3} n^{2}}{4}-\frac {i \pi \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{3} n^{2}}{4}+\frac {i \pi \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} n^{2}}{2}-\frac {i \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \pi \,\operatorname {csgn}\left (i \left (a x -1\right )\right )}{2}-\frac {i \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i \left (a x +1\right )\right ) \pi }{2}-\frac {i \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \pi }{2}-\frac {i \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i c \right ) \pi }{2}-\frac {i \pi \operatorname {csgn}\left (i \left (a x -1\right )\right )^{3} n}{2}+\frac {i \pi \operatorname {csgn}\left (i \left (a x -1\right )\right )^{2} n}{2}-\frac {i n \pi }{2}+i \pi \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3}}{2}-\frac {i n^{2} \pi }{2}}}{a^{3} \left (n^{2}-1\right ) n}\)	\(688\)

Input:

int(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x,method=_RETURNVE 
RBOSE)
 

Output:

-(a*x-1)*(a*x+1)*(a*n*x-1)*exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(-1-1/2*n^2) 
/a^3/n/(n^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int e^{n \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=-\frac {{\left (a^{3} n x^{3} - a^{2} x^{2} - a n x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{3} n^{3} - a^{3} n} \] Input:

integrate(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm 
="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.36 \[ \int e^{n \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {{\left (a n x - 1\right )} c^{-\frac {1}{2} \, n^{2} - 1} e^{\left (-\frac {1}{2} \, n^{2} \log \left (a x + 1\right ) - \frac {1}{2} \, n^{2} \log \left (-a x + 1\right ) + \frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (-a x + 1\right )\right )}}{{\left (n^{3} - n\right )} a^{3}} \] Input:

integrate(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm 
="maxima")
 

Output:

(a*n*x - 1)*c^(-1/2*n^2 - 1)*e^(-1/2*n^2*log(a*x + 1) - 1/2*n^2*log(-a*x + 
 1) + 1/2*n*log(a*x + 1) - 1/2*n*log(-a*x + 1))/((n^3 - n)*a^3)
 

Giac [F]

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

integrate(exp(n*arctanh(a*x))*x^2*(-a^2*c*x^2+c)^(-1-1/2*n^2),x, algorithm 
="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 13.76 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.91 \[ \int e^{n \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{-1-\frac {n^2}{2}} \, dx=\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}-a\,n\,x\,{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}}{a^3\,c\,n\,{\left (c-a^2\,c\,x^2\right )}^{\frac {n^2}{2}}-a^3\,c\,n^3\,{\left (c-a^2\,c\,x^2\right )}^{\frac {n^2}{2}}} \] Input:

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

Output:

(exp((n*log(a*x + 1))/2 - (n*log(1 - a*x))/2) - a*n*x*exp((n*log(a*x + 1)) 
/2 - (n*log(1 - a*x))/2))/(a^3*c*n*(c - a^2*c*x^2)^(n^2/2) - a^3*c*n^3*(c 
- a^2*c*x^2)^(n^2/2))
 

Reduce [F]

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

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

Output:

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