\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^{9/8}} \, dx\) [1330]

Optimal result

Integrand size = 29, antiderivative size = 79 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\frac {4 \sqrt [8]{2} a \sqrt [8]{1-a^2 x^2} \operatorname {AppellF1}\left (-\frac {3}{8},\frac {7}{8},2,\frac {5}{8},\frac {1}{2} (1-a x),1-a x\right )}{3 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \] Output:

4/3*2^(1/8)*a*(-a^2*x^2+1)^(1/8)*AppellF1(-3/8,7/8,2,5/8,-1/2*a*x+1/2,-a*x 
+1)/c/(-a*x+1)^(3/8)/(-a^2*c*x^2+c)^(1/8)
 

Mathematica [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx \] Input:

Integrate[E^(ArcTanh[a*x]/2)/(x^2*(c - a^2*c*x^2)^(9/8)),x]
 

Output:

Integrate[E^(ArcTanh[a*x]/2)/(x^2*(c - a^2*c*x^2)^(9/8)), x]
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6703, 6700, 153}

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.

\(\displaystyle \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \frac {\sqrt [8]{1-a^2 x^2} \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (1-a^2 x^2\right )^{9/8}}dx}{c \sqrt [8]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\sqrt [8]{1-a^2 x^2} \int \frac {1}{x^2 (1-a x)^{11/8} (a x+1)^{7/8}}dx}{c \sqrt [8]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 153

\(\displaystyle \frac {2\ 2^{5/8} a \sqrt [8]{a x+1} \sqrt [8]{1-a^2 x^2} \operatorname {AppellF1}\left (\frac {1}{8},\frac {11}{8},2,\frac {9}{8},\frac {1}{2} (a x+1),a x+1\right )}{c \sqrt [8]{c-a^2 c x^2}}\)

Input:

Int[E^(ArcTanh[a*x]/2)/(x^2*(c - a^2*c*x^2)^(9/8)),x]
 

Output:

(2*2^(5/8)*a*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8)*AppellF1[1/8, 11/8, 2, 9/ 
8, (1 + a*x)/2, 1 + a*x])/(c*(c - a^2*c*x^2)^(1/8))
 

Defintions of rubi rules used

rule 153

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp 
lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c 
 - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( 
b*c - a*d)], 0] &&  !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, 
a + b*x])
 

rule 6700

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

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

Maple [F]

\[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{8}}}d x\]

Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x)
 

Output:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x, a 
lgorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\text {Timed out} \] Input:

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2/(-a**2*c*x**2+c)**(9 
/8),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}} x^{2}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x, a 
lgorithm="maxima")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(9/8)*x^2), 
 x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{8}} x^{2}} \,d x } \] Input:

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x, a 
lgorithm="giac")
 

Output:

integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/((-a^2*c*x^2 + c)^(9/8)*x^2), 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\int \frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{9/8}} \,d x \] Input:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x^2*(c - a^2*c*x^2)^(9/8)),x)
 

Output:

int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(x^2*(c - a^2*c*x^2)^(9/8)), x)
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{9/8}} \, dx=\frac {-52 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {7}{8}}+648 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {7}{8}} a^{3} x^{3}+36 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {7}{8}} a^{2} x^{2}-648 \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {7}{8}} a x +405 \left (-a^{2} x^{2}+1\right )^{\frac {5}{4}} \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} a^{2} x^{2}-\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}}}d x \right ) a^{4} x +18 \left (-a^{2} x^{2}+1\right )^{\frac {5}{4}} \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} a^{2} x^{2}-\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}}}d x \right ) a^{3} x -26 \left (-a^{2} x^{2}+1\right )^{\frac {5}{4}} \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} a^{2} x^{3}-\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} x}d x \right ) a x -477 \left (-a^{2} x^{2}+1\right )^{\frac {5}{4}} \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{4}}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} a^{2} x^{2}-\left (-a^{2} x^{2}+1\right )^{\frac {5}{8}}}d x \right ) a^{2} x}{52 c^{\frac {5}{8}} \sqrt {c}\, \left (-a^{2} x^{2}+1\right )^{\frac {5}{4}} x} \] Input:

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/x^2/(-a^2*c*x^2+c)^(9/8),x)
 

Output:

(c**(3/8)*( - 16*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(7/8) + 648*sqrt(a*x + 
1)*( - a**2*x**2 + 1)**(7/8)*a**3*x**3 + 36*sqrt(a*x + 1)*( - a**2*x**2 + 
1)**(7/8)*a**2*x**2 - 648*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(7/8)*a*x - 36 
*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(7/8) + 405*( - a**2*x**2 + 1)**(5/4)*i 
nt((sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4)*x**2)/(( - a**2*x**2 + 1)**(5/ 
8)*a**2*x**2 - ( - a**2*x**2 + 1)**(5/8)),x)*a**4*x + 18*( - a**2*x**2 + 1 
)**(5/4)*int((sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4)*x)/(( - a**2*x**2 + 
1)**(5/8)*a**2*x**2 - ( - a**2*x**2 + 1)**(5/8)),x)*a**3*x - 26*( - a**2*x 
**2 + 1)**(5/4)*int((sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4))/(( - a**2*x* 
*2 + 1)**(5/8)*a**2*x**3 - ( - a**2*x**2 + 1)**(5/8)*x),x)*a*x - 477*( - a 
**2*x**2 + 1)**(5/4)*int((sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4))/(( - a* 
*2*x**2 + 1)**(5/8)*a**2*x**2 - ( - a**2*x**2 + 1)**(5/8)),x)*a**2*x))/(52 
*sqrt(c)*( - a**2*x**2 + 1)**(5/4)*c*x)