\(\int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx\) [275]

Optimal result

Integrand size = 20, antiderivative size = 78 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-n),-\frac {n}{2},\frac {1-n}{2},\frac {1}{2} (1-a x)\right )}{a c (1+n) \sqrt {c-a c x}} \] Output:

2^(1+1/2*n)*hypergeom([-1/2*n, -1/2-1/2*n],[1/2-1/2*n],-1/2*a*x+1/2)/a/c/( 
1+n)/((-a*x+1)^(1/2*n))/(-a*c*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-\frac {n}{2},-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )}{a c (1+n) \sqrt {c-a c x}} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(c - a*c*x)^(3/2),x]
 

Output:

(2^(1 + n/2)*Hypergeometric2F1[-1/2 - n/2, -1/2*n, 1/2 - n/2, 1/2 - (a*x)/ 
2])/(a*c*(1 + n)*(1 - a*x)^(n/2)*Sqrt[c - a*c*x])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6680, 37, 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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{(c-a c x)^{3/2}}dx\)

\(\Big \downarrow \) 37

\(\displaystyle \frac {(1-a x)^{3/2} \int (1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{n/2}dx}{(c-a c x)^{3/2}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{\frac {n}{2}+1} (1-a x)^{\frac {1}{2} (-n-1)+\frac {3}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),-\frac {n}{2},\frac {1-n}{2},\frac {1}{2} (1-a x)\right )}{a (n+1) (c-a c x)^{3/2}}\)

Input:

Int[E^(n*ArcTanh[a*x])/(c - a*c*x)^(3/2),x]
 

Output:

(2^(1 + n/2)*(1 - a*x)^(3/2 + (-1 - n)/2)*Hypergeometric2F1[(-1 - n)/2, -1 
/2*n, (1 - n)/2, (1 - a*x)/2])/(a*(1 + n)*(c - a*c*x)^(3/2))
 

Defintions of rubi rules used

rule 37

Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S 
imp[(a + b*x)^m/(c + d*x)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !SimplerQ[a + b*x, c + d*x]
 

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 6680

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {3}{2}}}d x\]

Input:

int(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x)
 

Output:

int(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-a*c*x + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*x^2 - 2* 
a*c^2*x + c^2), x)
 

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(exp(n*atanh(a*x))/(-a*c*x+c)**(3/2),x)
 

Output:

Integral(exp(n*atanh(a*x))/(-c*(a*x - 1))**(3/2), x)
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(-a*c*x + c)^(3/2), x)
 

Giac [F]

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

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(-a*c*x + c)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{3/2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(c - a*c*x)^(3/2),x)
 

Output:

int(exp(n*atanh(a*x))/(c - a*c*x)^(3/2), x)
 

Reduce [F]

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

int(exp(n*atanh(a*x))/(-a*c*x+c)^(3/2),x)
 

Output:

( - int(e**(atanh(a*x)*n)/(sqrt( - a*x + 1)*a*x - sqrt( - a*x + 1)),x))/(s 
qrt(c)*c)