\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{(c-a^2 c x^2)^{9/2}} \, dx\) [1318]

Optimal result

Integrand size = 26, antiderivative size = 165 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {2 e^{\frac {1}{2} \text {arctanh}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac {112 e^{\frac {1}{2} \text {arctanh}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {256 e^{\frac {1}{2} \text {arctanh}(a x)} (1-6 a x)}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2048 e^{\frac {1}{2} \text {arctanh}(a x)} (1-2 a x)}{6435 a c^4 \sqrt {c-a^2 c x^2}} \] Output:

-2/195*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-14*a*x+1)/a/c/(-a^2*c*x^2+c)^( 
7/2)-112/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-10*a*x+1)/a/c^2/(-a^2*c 
*x^2+c)^(5/2)-256/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-6*a*x+1)/a/c^3 
/(-a^2*c*x^2+c)^(3/2)-2048/6435*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-2*a*x 
+1)/a/c^4/(-a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \left (1241-3838 a x-3384 a^2 x^2+8240 a^3 x^3+3200 a^4 x^4-6912 a^5 x^5-1024 a^6 x^6+2048 a^7 x^7\right )}{6435 a c^4 (1-a x)^{15/4} (1+a x)^{13/4} \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

(-2*Sqrt[1 - a^2*x^2]*(1241 - 3838*a*x - 3384*a^2*x^2 + 8240*a^3*x^3 + 320 
0*a^4*x^4 - 6912*a^5*x^5 - 1024*a^6*x^6 + 2048*a^7*x^7))/(6435*a*c^4*(1 - 
a*x)^(15/4)*(1 + a*x)^(13/4)*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.

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

\(\Big \downarrow \) 6686

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

\(\Big \downarrow \) 6686

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

\(\Big \downarrow \) 6686

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

\(\Big \downarrow \) 6685

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

Input:

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

Output:

(-2*E^(ArcTanh[a*x]/2)*(1 - 14*a*x))/(195*a*c*(c - a^2*c*x^2)^(7/2)) + (56 
*((-2*E^(ArcTanh[a*x]/2)*(1 - 10*a*x))/(99*a*c*(c - a^2*c*x^2)^(5/2)) + (8 
0*((-2*E^(ArcTanh[a*x]/2)*(1 - 6*a*x))/(35*a*c*(c - a^2*c*x^2)^(3/2)) - (1 
6*E^(ArcTanh[a*x]/2)*(1 - 2*a*x))/(35*a*c^2*Sqrt[c - a^2*c*x^2])))/(99*c)) 
)/(65*c)
 

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 = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62

method	result	size
gosper	\(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 a^{5} x^{5}+3200 a^{4} x^{4}+8240 a^{3} x^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\)	\(103\)
orering	\(\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 a^{5} x^{5}+3200 a^{4} x^{4}+8240 a^{3} x^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\)	\(103\)

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/2),x,method=_RETU 
RNVERBOSE)
 

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*c*x^2+c)^(9/2)
 

Fricas [F(-1)]

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

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/2),x, algor 
ithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/2),x, algor 
ithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

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

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(9/2),x, algor 
ithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 26.89 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {2482}{6435\,a^7\,c^4}+\frac {4096\,x^7}{6435\,c^4}-\frac {7676\,x}{6435\,a^6\,c^4}-\frac {2048\,x^6}{6435\,a\,c^4}-\frac {1536\,x^5}{715\,a^2\,c^4}+\frac {1280\,x^4}{1287\,a^3\,c^4}+\frac {3296\,x^3}{1287\,a^4\,c^4}-\frac {752\,x^2}{715\,a^5\,c^4}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {2 \sqrt {c}\, \left (-a x +1\right )^{\frac {1}{4}} \left (-2048 a^{7} x^{7}+1024 a^{6} x^{6}+6912 a^{5} x^{5}-3200 a^{4} x^{4}-8240 a^{3} x^{3}+3384 a^{2} x^{2}+3838 a x -1241\right )}{6435 \left (a x +1\right )^{\frac {1}{4}} a \,c^{5} \left (a^{7} x^{7}-a^{6} x^{6}-3 a^{5} x^{5}+3 a^{4} x^{4}+3 a^{3} x^{3}-3 a^{2} x^{2}-a x +1\right )} \] Input:

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

Output:

(2*sqrt(c)*( - 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*c**5*(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))