\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) [297]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {7+n}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7+n}{2},\frac {7}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{5 (c-a c x)^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left ((9+2 n-3 a x) (1+a x)+3 (-1+a x)^2 \left (\frac {-1+a x}{1+a x}\right )^{\frac {1+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {3}{2},\frac {2}{1+a x}\right )\right )}{2 a c^3 (3+n) (5+n) (-1+a x)^2 \sqrt {c-a c x}} \] Input:

Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]
 

Output:

((1 + 1/(a*x))^(n/2)*((9 + 2*n - 3*a*x)*(1 + a*x) + 3*(-1 + a*x)^2*((-1 + 
a*x)/(1 + a*x))^((1 + n)/2)*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + 
a*x)]))/(2*a*c^3*(3 + n)*(5 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x)^2*Sqrt[c - 
 a*c*x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(98)=196\).

Time = 0.61 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.54, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6727, 105, 105, 142}

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.

\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6727

\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-7)} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{3/2}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {a \left (\frac {1}{x}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+5}-\frac {3 a \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{n/2} \sqrt {\frac {1}{x}}d\frac {1}{x}}{2 (n+5)}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {a \left (\frac {1}{x}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+5}-\frac {3 a \left (\frac {a \sqrt {\frac {1}{x}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+3}-\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (1+\frac {1}{a x}\right )^{n/2}}{\sqrt {\frac {1}{x}}}d\frac {1}{x}}{2 (n+3)}\right )}{2 (n+5)}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 142

\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {a \left (\frac {1}{x}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+5}-\frac {3 a \left (\frac {a \sqrt {\frac {1}{x}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+3}-\frac {a \sqrt {\frac {1}{x}} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2},\frac {3}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{n+3}\right )}{2 (n+5)}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

Input:

Int[E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]
 

Output:

-(((1 - 1/(a*x))^(7/2)*((a*(1 - 1/(a*x))^((-5 - n)/2)*(1 + 1/(a*x))^((2 + 
n)/2)*(x^(-1))^(3/2))/(5 + n) - (3*a*((a*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1 
/(a*x))^((2 + n)/2)*Sqrt[x^(-1)])/(3 + n) - (a*((a - x^(-1))/(a + x^(-1))) 
^((3 + n)/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*Sqrt[x^( 
-1)]*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/((a + x^(-1))*x)])/(3 + n))) 
/(2*(5 + n))))/((x^(-1))^(7/2)*(c - a*c*x)^(7/2)))
 

Defintions of rubi rules used

rule 105

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

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

rule 6727

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (-a c x +c \right )^{\frac {7}{2}}}d x\]

Input:

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x)
 

Output:

int(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-a*c*x + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^4*x^4 - 4*a 
^3*c^4*x^3 + 6*a^2*c^4*x^2 - 4*a*c^4*x + c^4), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(exp(n*acoth(a*x))/(-a*c*x+c)**(7/2),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="maxima")
 

Output:

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(-a*c*x + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a c x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/(-a*c*x+c)^(7/2),x, algorithm="giac")
 

Output:

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(-a*c*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a\,c\,x\right )}^{7/2}} \,d x \] Input:

int(exp(n*acoth(a*x))/(c - a*c*x)^(7/2),x)
 

Output:

int(exp(n*acoth(a*x))/(c - a*c*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{\sqrt {-a x +1}\, a^{3} x^{3}-3 \sqrt {-a x +1}\, a^{2} x^{2}+3 \sqrt {-a x +1}\, a x -\sqrt {-a x +1}}d x}{\sqrt {c}\, c^{3}} \] Input:

int(exp(n*acoth(a*x))/(-a*c*x+c)^(7/2),x)
 

Output:

( - int(e**(acoth(a*x)*n)/(sqrt( - a*x + 1)*a**3*x**3 - 3*sqrt( - a*x + 1) 
*a**2*x**2 + 3*sqrt( - a*x + 1)*a*x - sqrt( - a*x + 1)),x))/(sqrt(c)*c**3)