\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{(1-a^2 x^2)^{9/2}} \, dx\) [1310]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {2 e^{\frac {1}{2} \text {arctanh}(a x)} (1-14 a x)}{195 a \left (1-a^2 x^2\right )^{7/2}}-\frac {112 e^{\frac {1}{2} \text {arctanh}(a x)} (1-10 a x)}{6435 a \left (1-a^2 x^2\right )^{5/2}}-\frac {256 e^{\frac {1}{2} \text {arctanh}(a x)} (1-6 a x)}{6435 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2048 e^{\frac {1}{2} \text {arctanh}(a x)} (1-2 a x)}{6435 a \sqrt {1-a^2 x^2}} \] Output:

-2/195*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-14*a*x+1)/a/(-a^2*x^2+1)^(7/2) 
-112/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-10*a*x+1)/a/(-a^2*x^2+1)^(5 
/2)-256/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-6*a*x+1)/a/(-a^2*x^2+1)^ 
(3/2)-2048/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-2*a*x+1)/a/(-a^2*x^2+ 
1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {2 \left (1241-3838 a x-3384 a^2 x^2+8240 a^3 x^3+3200 a^4 x^4-6912 a^5 x^5-1024 a^6 x^6+2048 a^7 x^7\right )}{6435 a (1-a x)^{15/4} (1+a x)^{13/4}} \] Input:

Integrate[E^(ArcTanh[a*x]/2)/(1 - a^2*x^2)^(9/2),x]
 

Output:

(-2*(1241 - 3838*a*x - 3384*a^2*x^2 + 8240*a^3*x^3 + 3200*a^4*x^4 - 6912*a 
^5*x^5 - 1024*a^6*x^6 + 2048*a^7*x^7))/(6435*a*(1 - a*x)^(15/4)*(1 + a*x)^ 
(13/4))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6686, 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.

\(\displaystyle \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {56}{65} \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{7/2}}dx-\frac {2 (1-14 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {56}{65} \left (\frac {80}{99} \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{5/2}}dx-\frac {2 (1-10 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{99 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {2 (1-14 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6686

\(\displaystyle \frac {56}{65} \left (\frac {80}{99} \left (\frac {24}{35} \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {2 (1-6 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {2 (1-10 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{99 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {2 (1-14 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}\)

\(\Big \downarrow \) 6685

\(\displaystyle \frac {56}{65} \left (\frac {80}{99} \left (-\frac {2 (1-6 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 (1-2 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{35 a \sqrt {1-a^2 x^2}}\right )-\frac {2 (1-10 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{99 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {2 (1-14 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{195 a \left (1-a^2 x^2\right )^{7/2}}\)

Input:

Int[E^(ArcTanh[a*x]/2)/(1 - a^2*x^2)^(9/2),x]
 

Output:

(-2*E^(ArcTanh[a*x]/2)*(1 - 14*a*x))/(195*a*(1 - a^2*x^2)^(7/2)) + (56*((- 
2*E^(ArcTanh[a*x]/2)*(1 - 10*a*x))/(99*a*(1 - a^2*x^2)^(5/2)) + (80*((-2*E 
^(ArcTanh[a*x]/2)*(1 - 6*a*x))/(35*a*(1 - a^2*x^2)^(3/2)) - (16*E^(ArcTanh 
[a*x]/2)*(1 - 2*a*x))/(35*a*Sqrt[1 - a^2*x^2])))/99))/65
 

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 = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68

method	result	size
gosper	\(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 a^{5} x^{5}+3200 a^{4} x^{4}+8240 a^{3} x^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\)	\(102\)
orering	\(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 a^{5} x^{5}+3200 a^{4} x^{4}+8240 a^{3} x^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\)	\(102\)

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x,method=_RETURN 
VERBOSE)
 

Output:

2/6435*(a*x-1)*(a*x+1)*(2048*a^7*x^7-1024*a^6*x^6-6912*a^5*x^5+3200*a^4*x^ 
4+8240*a^3*x^3-3384*a^2*x^2-3838*a*x+1241)*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1 
/2)/a/(-a^2*x^2+1)^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (2048 \, a^{7} x^{7} - 1024 \, a^{6} x^{6} - 6912 \, a^{5} x^{5} + 3200 \, a^{4} x^{4} + 8240 \, a^{3} x^{3} - 3384 \, a^{2} x^{2} - 3838 \, a x + 1241\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{6435 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x, algorit 
hm="fricas")
 

Output:

-2/6435*(2048*a^7*x^7 - 1024*a^6*x^6 - 6912*a^5*x^5 + 3200*a^4*x^4 + 8240* 
a^3*x^3 - 3384*a^2*x^2 - 3838*a*x + 1241)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a 
^2*x^2 + 1)/(a*x - 1))/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + a)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x, algorit 
hm="maxima")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/(-a^2*x^2 + 1)^(9/2), x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x, algorit 
hm="giac")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/(-a^2*x^2 + 1)^(9/2), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 27.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {2482\,\sqrt {1-a^2\,x^2}}{6435\,a^9}-\frac {7676\,x\,\sqrt {1-a^2\,x^2}}{6435\,a^8}+\frac {4096\,x^7\,\sqrt {1-a^2\,x^2}}{6435\,a^2}-\frac {2048\,x^6\,\sqrt {1-a^2\,x^2}}{6435\,a^3}-\frac {1536\,x^5\,\sqrt {1-a^2\,x^2}}{715\,a^4}+\frac {1280\,x^4\,\sqrt {1-a^2\,x^2}}{1287\,a^5}+\frac {3296\,x^3\,\sqrt {1-a^2\,x^2}}{1287\,a^6}-\frac {752\,x^2\,\sqrt {1-a^2\,x^2}}{715\,a^7}\right )}{\frac {1}{a^8}+x^8-\frac {4\,x^6}{a^2}+\frac {6\,x^4}{a^4}-\frac {4\,x^2}{a^6}} \] Input:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(1 - a^2*x^2)^(9/2),x)
 

Output:

-(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)*((2482*(1 - a^2*x^2)^(1/2))/(6435* 
a^9) - (7676*x*(1 - a^2*x^2)^(1/2))/(6435*a^8) + (4096*x^7*(1 - a^2*x^2)^( 
1/2))/(6435*a^2) - (2048*x^6*(1 - a^2*x^2)^(1/2))/(6435*a^3) - (1536*x^5*( 
1 - a^2*x^2)^(1/2))/(715*a^4) + (1280*x^4*(1 - a^2*x^2)^(1/2))/(1287*a^5) 
+ (3296*x^3*(1 - a^2*x^2)^(1/2))/(1287*a^6) - (752*x^2*(1 - a^2*x^2)^(1/2) 
)/(715*a^7)))/(1/a^8 + x^8 - (4*x^6)/a^2 + (6*x^4)/a^4 - (4*x^2)/a^6)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {2 \left (-a x +1\right )^{\frac {1}{4}} \left (-2048 a^{7} x^{7}+1024 a^{6} x^{6}+6912 a^{5} x^{5}-3200 a^{4} x^{4}-8240 a^{3} x^{3}+3384 a^{2} x^{2}+3838 a x -1241\right )}{6435 \left (a x +1\right )^{\frac {1}{4}} a \left (a^{7} x^{7}-a^{6} x^{6}-3 a^{5} x^{5}+3 a^{4} x^{4}+3 a^{3} x^{3}-3 a^{2} x^{2}-a x +1\right )} \] Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*x^2+1)^(9/2),x)
 

Output:

(2*( - a*x + 1)**(1/4)*( - 2048*a**7*x**7 + 1024*a**6*x**6 + 6912*a**5*x** 
5 - 3200*a**4*x**4 - 8240*a**3*x**3 + 3384*a**2*x**2 + 3838*a*x - 1241))/( 
6435*(a*x + 1)**(1/4)*a*(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 
 + 3*a**3*x**3 - 3*a**2*x**2 - a*x + 1))