\(\int \frac {e^{n \coth ^{-1}(a x)}}{x (c-a^2 c x^2)^{5/2}} \, dx\) [735]

Optimal result

Integrand size = 27, antiderivative size = 944 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Output:

-a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5 
/(3+n)/(-a^2*c*x^2+c)^(5/2)-3*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2* 
n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)+6*a^5*(1-1/ 
a^2/x^2)^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(3+n)/(-n^ 
2+1)/(-a^2*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(3/2-1/2*n)* 
(1+1/a/x)^(-3/2+1/2*n)*x^5/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(5/2)+4*a^5*(1-1/ 
a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^5/(3+n)/(-a 
^2*c*x^2+c)^(5/2)+8*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2*n)*(1+1/a/ 
x)^(-1/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)-8*a^5*(1-1/a^2/x^2)^( 
5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^5/(3+n)/(-n^2+1)/(-a^2 
*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x) 
^(1/2+1/2*n)*x^5/(3+n)/(-a^2*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)*(1-1 
/a/x)^(-1/2-1/2*n)*(1+1/a/x)^(1/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5 
/2)+4*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(3/2+1/2*n) 
*x^5/(3+n)/(-a^2*c*x^2+c)^(5/2)-2^(5/2+1/2*n)*a^5*(1-1/a^2/x^2)^(5/2)*(1-1 
/a/x)^(-3/2-1/2*n)*x^5*hypergeom([-3/2-1/2*n, -3/2-1/2*n],[-1/2-1/2*n],1/2 
*(a-1/x)/a)/(3+n)/(-a^2*c*x^2+c)^(5/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.23 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (42-2 n^2-45 a n x+5 a n^3 x\right )+6 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (-1+n^2\right ) \left (-1+a^2 x^2\right ) \cosh \left (3 \coth ^{-1}(a x)\right )\right )-8 e^{(1+n) \coth ^{-1}(a x)} \left (9-9 n-n^2+n^3\right ) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (-1+n) (1+n) \left (-9+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \] Input:

Integrate[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(5/2)),x]
 

Output:

(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(42 - 2*n^2 - 45*a*n*x + 5* 
a*n^3*x) + 6*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[2*ArcCoth[a*x]] - n 
*(-1 + n^2)*(-1 + a^2*x^2)*Cosh[3*ArcCoth[a*x]]) - 8*E^((1 + n)*ArcCoth[a* 
x])*(9 - 9*n - n^2 + n^3)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, ( 
3 + n)/2, -E^(2*ArcCoth[a*x])])/(4*a*c^2*(-1 + n)*(1 + n)*(-9 + n^2)*Sqrt[ 
1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.66, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6746, 6749, 137, 2009}

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.

\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6746

\(\displaystyle \frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^6}dx}{\left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6749

\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}}{x^4}d\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 137

\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \left (-4 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+1}+6 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+2}-4 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+3}+a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+4}+a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}\right )d\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {a^5 2^{\frac {n+5}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-3),\frac {1}{2} (-n-3),\frac {1}{2} (-n-1),\frac {a-\frac {1}{x}}{2 a}\right )}{n+3}+\frac {3 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}-\frac {8 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}-\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right )}+\frac {8 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right )}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{n^4-10 n^2+9}+\frac {a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}-\frac {4 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}-\frac {4 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\)

Input:

Int[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(5/2)),x]
 

Output:

-(((1 - 1/(a^2*x^2))^(5/2)*x^5*((a^5*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a* 
x))^((-3 + n)/2))/(3 + n) + (3*a^5*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x) 
)^((-3 + n)/2))/(3 + 4*n + n^2) - (6*a^5*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/ 
(a*x))^((-3 + n)/2))/((3 + n)*(1 - n^2)) + (6*a^5*(1 - 1/(a*x))^((3 - n)/2 
)*(1 + 1/(a*x))^((-3 + n)/2))/(9 - 10*n^2 + n^4) - (4*a^5*(1 - 1/(a*x))^(( 
-3 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/(3 + n) - (8*a^5*(1 - 1/(a*x))^((-1 
 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/(3 + 4*n + n^2) + (8*a^5*(1 - 1/(a*x) 
)^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/((3 + n)*(1 - n^2)) + (6*a^5*(1 
- 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((1 + n)/2))/(3 + n) + (6*a^5*(1 - 1 
/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2))/(3 + 4*n + n^2) - (4*a^5*( 
1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((3 + n)/2))/(3 + n) + (2^((5 + n) 
/2)*a^5*(1 - 1/(a*x))^((-3 - n)/2)*Hypergeometric2F1[(-3 - n)/2, (-3 - n)/ 
2, (-1 - n)/2, (a - x^(-1))/(2*a)])/(3 + n)))/(c - a^2*c*x^2)^(5/2))
 

Defintions of rubi rules used

rule 137

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && 
ILtQ[n, 0]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6746

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

rule 6749

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

Input:

int(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

int(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas" 
)
 

Output:

integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^7 
- 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 - c^3*x), x)
 

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(exp(n*acoth(a*x))/x/(-a**2*c*x**2+c)**(5/2),x)
 

Output:

Integral(exp(n*acoth(a*x))/(x*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)
 

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima" 
)
 

Output:

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x), x)
 

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

Output:

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \] Input:

int(exp(n*acoth(a*x))/(x*(c - a^2*c*x^2)^(5/2)),x)
 

Output:

int(exp(n*acoth(a*x))/(x*(c - a^2*c*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{\sqrt {-a^{2} x^{2}+1}\, a^{4} x^{5}-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{3}+\sqrt {-a^{2} x^{2}+1}\, x}d x}{\sqrt {c}\, c^{2}} \] Input:

int(exp(n*acoth(a*x))/x/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

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