\(\int \frac {e^{\frac {1}{2} \text {arctanh}(a x)} x}{(c-a^2 c x^2)^{9/8}} \, dx\) [1342]

Optimal result

Integrand size = 27, antiderivative size = 131 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^{9/8}} \, dx=\frac {4 \sqrt [8]{1+a x} \sqrt [8]{1-a^2 x^2}}{a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}}-\frac {8 \sqrt [8]{2} \sqrt [8]{1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{8},\frac {7}{8},\frac {5}{8},\frac {1}{2} (1-a x)\right )}{3 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^{9/8}} \, dx=-\frac {4 \sqrt [8]{1-a^2 x^2} \left (-5 \sqrt [8]{1+a x}+2 \sqrt [8]{2} (-1+a x) \operatorname {Hypergeometric2F1}\left (\frac {5}{8},\frac {7}{8},\frac {13}{8},\frac {1}{2} (1-a x)\right )\right )}{15 a^2 c (1-a x)^{3/8} \sqrt [8]{c-a^2 c x^2}} \] Input:

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

Output:

(-4*(1 - a^2*x^2)^(1/8)*(-5*(1 + a*x)^(1/8) + 2*2^(1/8)*(-1 + a*x)*Hyperge 
ometric2F1[5/8, 7/8, 13/8, (1 - a*x)/2]))/(15*a^2*c*(1 - a*x)^(3/8)*(c - a 
^2*c*x^2)^(1/8))
 

Rubi [A] (warning: unable to verify)

Time = 0.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6703, 6700, 87, 79}

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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {\sqrt [8]{1-a^2 x^2} \left (\frac {4 \sqrt [8]{a x+1}}{3 a^2 (1-a x)^{3/8}}-\frac {2 \int \frac {1}{(1-a x)^{3/8} (a x+1)^{7/8}}dx}{3 a}\right )}{c \sqrt [8]{c-a^2 c x^2}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\sqrt [8]{1-a^2 x^2} \left (\frac {8 \sqrt [8]{2} (1-a x)^{5/8} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},\frac {7}{8},\frac {13}{8},\frac {1}{2} (1-a x)\right )}{15 a^2}+\frac {4 \sqrt [8]{a x+1}}{3 a^2 (1-a x)^{3/8}}\right )}{c \sqrt [8]{c-a^2 c x^2}}\)

Input:

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

Output:

((1 - a^2*x^2)^(1/8)*((4*(1 + a*x)^(1/8))/(3*a^2*(1 - a*x)^(3/8)) + (8*2^( 
1/8)*(1 - a*x)^(5/8)*Hypergeometric2F1[5/8, 7/8, 13/8, (1 - a*x)/2])/(15*a 
^2)))/(c*(c - a^2*c*x^2)^(1/8))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^{9/8}} \, dx=\frac {-4 c \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{8}} a x +8 c \sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {1}{8}}-3 c \sqrt {-a^{2} x^{2}+1}\, \left (\int \frac {\sqrt {a x +1}\, \left (-a^{2} x^{2}+1\right )^{\frac {5}{8}} x}{a^{3} x^{3}+a^{2} x^{2}-a x -1}d x \right ) a^{2}}{c^{\frac {17}{8}} \sqrt {-a^{2} x^{2}+1}\, a^{2}} \] Input:

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

Output:

(c**(1/8)*( - 4*c*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/8)*a*x + 4*c*sqrt(a 
*x + 1)*( - a**2*x**2 + 1)**(1/8) + 4*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1 
/8)*c - 3*c*sqrt( - a**2*x**2 + 1)*int((sqrt(a*x + 1)*( - a**2*x**2 + 1)** 
(5/8)*x)/(a**3*x**3 + a**2*x**2 - a*x - 1),x)*a**2))/(c**(1/4)*sqrt( - a** 
2*x**2 + 1)*a**2*c**2)