\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\) [1393]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {e^{n \text {arctanh}(a x)} (n-5 a x)}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {20 e^{n \text {arctanh}(a x)} (n-3 a x)}{a c^2 \left (9-n^2\right ) \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {120 e^{n \text {arctanh}(a x)} (n-a x)}{a c^3 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \sqrt {c-a^2 c x^2}} \] Output:

-exp(n*arctanh(a*x))*(-5*a*x+n)/a/c/(-n^2+25)/(-a^2*c*x^2+c)^(5/2)-20*exp( 
n*arctanh(a*x))*(-3*a*x+n)/a/c^2/(-n^2+9)/(-n^2+25)/(-a^2*c*x^2+c)^(3/2)-1 
20*exp(n*arctanh(a*x))*(-a*x+n)/a/c^3/(-n^2+1)/(-n^2+9)/(-n^2+25)/(-a^2*c* 
x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {(1-a x)^{\frac {1}{2} (-5-n)} (1+a x)^{\frac {1}{2} (-5+n)} \sqrt {1-a^2 x^2} \left (-n^5+5 a n^4 x+n^3 \left (30-20 a^2 x^2\right )+10 a n^2 x \left (-11+6 a^2 x^2\right )+n \left (-149+260 a^2 x^2-120 a^4 x^4\right )+15 a x \left (15-20 a^2 x^2+8 a^4 x^4\right )\right )}{a c^3 (-5+n) (-3+n) (-1+n) (1+n) (3+n) (5+n) \sqrt {c-a^2 c x^2}} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]
 

Output:

-(((1 - a*x)^((-5 - n)/2)*(1 + a*x)^((-5 + n)/2)*Sqrt[1 - a^2*x^2]*(-n^5 + 
 5*a*n^4*x + n^3*(30 - 20*a^2*x^2) + 10*a*n^2*x*(-11 + 6*a^2*x^2) + n*(-14 
9 + 260*a^2*x^2 - 120*a^4*x^4) + 15*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)))/(a 
*c^3*(-5 + n)*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(5 + n)*Sqrt[c - a^2*c*x^2 
]))
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6686, 6686, 6685}

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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {20 \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}}dx}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \text {arctanh}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {20 \left (\frac {6 \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}}dx}{c \left (9-n^2\right )}-\frac {(n-3 a x) e^{n \text {arctanh}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}\right )}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \text {arctanh}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6685

\(\displaystyle \frac {20 \left (-\frac {6 (n-a x) e^{n \text {arctanh}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \text {arctanh}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}\right )}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \text {arctanh}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\)

Input:

Int[E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]
 

Output:

-((E^(n*ArcTanh[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) 
 + (20*(-((E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^ 
(3/2))) - (6*E^(n*ArcTanh[a*x])*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*Sqrt 
[c - a^2*c*x^2])))/(c*(25 - n^2))
 

Defintions of rubi rules used

rule 6685

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

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]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (120 a^{5} x^{5}-120 n \,a^{4} x^{4}+60 a^{3} n^{2} x^{3}-20 a^{2} n^{3} x^{2}-300 a^{3} x^{3}+5 a \,n^{4} x +260 n \,x^{2} a^{2}-n^{5}-110 n^{2} x a +30 n^{3}+225 a x -149 n \right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a \left (n^{6}-35 n^{4}+259 n^{2}-225\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\)	\(140\)
orering	\(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (120 a^{5} x^{5}-120 n \,a^{4} x^{4}+60 a^{3} n^{2} x^{3}-20 a^{2} n^{3} x^{2}-300 a^{3} x^{3}+5 a \,n^{4} x +260 n \,x^{2} a^{2}-n^{5}-110 n^{2} x a +30 n^{3}+225 a x -149 n \right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a \left (n^{6}-35 n^{4}+259 n^{2}-225\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\)	\(140\)

Input:

int(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

(a*x-1)*(a*x+1)*(120*a^5*x^5-120*a^4*n*x^4+60*a^3*n^2*x^3-20*a^2*n^3*x^2-3 
00*a^3*x^3+5*a*n^4*x+260*a^2*n*x^2-n^5-110*a*n^2*x+30*n^3+225*a*x-149*n)*e 
xp(n*arctanh(a*x))/a/(n^6-35*n^4+259*n^2-225)/(-a^2*c*x^2+c)^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.76 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (120 \, a^{5} x^{5} - 120 \, a^{4} n x^{4} - n^{5} + 60 \, {\left (a^{3} n^{2} - 5 \, a^{3}\right )} x^{3} + 30 \, n^{3} - 20 \, {\left (a^{2} n^{3} - 13 \, a^{2} n\right )} x^{2} + 5 \, {\left (a n^{4} - 22 \, a n^{2} + 45 \, a\right )} x - 149 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{6} - 35 \, a c^{4} n^{4} + 259 \, a c^{4} n^{2} - {\left (a^{7} c^{4} n^{6} - 35 \, a^{7} c^{4} n^{4} + 259 \, a^{7} c^{4} n^{2} - 225 \, a^{7} c^{4}\right )} x^{6} - 225 \, a c^{4} + 3 \, {\left (a^{5} c^{4} n^{6} - 35 \, a^{5} c^{4} n^{4} + 259 \, a^{5} c^{4} n^{2} - 225 \, a^{5} c^{4}\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{6} - 35 \, a^{3} c^{4} n^{4} + 259 \, a^{3} c^{4} n^{2} - 225 \, a^{3} c^{4}\right )} x^{2}} \] Input:

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
 

Output:

-(120*a^5*x^5 - 120*a^4*n*x^4 - n^5 + 60*(a^3*n^2 - 5*a^3)*x^3 + 30*n^3 - 
20*(a^2*n^3 - 13*a^2*n)*x^2 + 5*(a*n^4 - 22*a*n^2 + 45*a)*x - 149*n)*sqrt( 
-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^4*n^6 - 35*a*c^4*n^4 + 
 259*a*c^4*n^2 - (a^7*c^4*n^6 - 35*a^7*c^4*n^4 + 259*a^7*c^4*n^2 - 225*a^7 
*c^4)*x^6 - 225*a*c^4 + 3*(a^5*c^4*n^6 - 35*a^5*c^4*n^4 + 259*a^5*c^4*n^2 
- 225*a^5*c^4)*x^4 - 3*(a^3*c^4*n^6 - 35*a^3*c^4*n^4 + 259*a^3*c^4*n^2 - 2 
25*a^3*c^4)*x^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(-a^2*c*x^2 + c)^(7/2), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 14.15 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.66 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {120\,x^5}{c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {120\,n\,x^4}{a\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}+\frac {x^3\,\left (60\,n^2-300\right )}{a^2\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {n\,\left (n^4-30\,n^2+149\right )}{a^5\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}+\frac {5\,x\,\left (n^4-22\,n^2+45\right )}{a^4\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}-\frac {20\,n\,x^2\,\left (n^2-13\right )}{a^3\,c^3\,\left (n^6-35\,n^4+259\,n^2-225\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^4}+x^4\,\sqrt {c-a^2\,c\,x^2}-\frac {2\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^2}\right )} \] Input:

int(exp(n*atanh(a*x))/(c - a^2*c*x^2)^(7/2),x)
 

Output:

-((a*x + 1)^(n/2)*((120*x^5)/(c^3*(259*n^2 - 35*n^4 + n^6 - 225)) - (120*n 
*x^4)/(a*c^3*(259*n^2 - 35*n^4 + n^6 - 225)) + (x^3*(60*n^2 - 300))/(a^2*c 
^3*(259*n^2 - 35*n^4 + n^6 - 225)) - (n*(n^4 - 30*n^2 + 149))/(a^5*c^3*(25 
9*n^2 - 35*n^4 + n^6 - 225)) + (5*x*(n^4 - 22*n^2 + 45))/(a^4*c^3*(259*n^2 
 - 35*n^4 + n^6 - 225)) - (20*n*x^2*(n^2 - 13))/(a^3*c^3*(259*n^2 - 35*n^4 
 + n^6 - 225))))/((1 - a*x)^(n/2)*((c - a^2*c*x^2)^(1/2)/a^4 + x^4*(c - a^ 
2*c*x^2)^(1/2) - (2*x^2*(c - a^2*c*x^2)^(1/2))/a^2))
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{\sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{3}} \] Input:

int(exp(n*atanh(a*x))/(-a^2*c*x^2+c)^(7/2),x)
 

Output:

( - int(e**(atanh(a*x)*n)/(sqrt( - a**2*x**2 + 1)*a**6*x**6 - 3*sqrt( - a* 
*2*x**2 + 1)*a**4*x**4 + 3*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2 
*x**2 + 1)),x))/(sqrt(c)*c**3)