3.8.55 \(\int \frac {e^{n \coth ^{-1}(a x)}}{x (c-a^2 c x^2)^{3/2}} \, dx\) [755]

3.8.55.1 Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {2^{\frac {1+n}{2}} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}} \]

output

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

3.8.55.2 Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.46 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a n x)-2 e^{\coth ^{-1}(a x)} (-1+n) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )}{a c (-1+n) (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \]

input

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

output

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

3.8.55.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6746, 6749, 100, 27, 88, 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.

\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6746

\(\displaystyle \frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4}dx}{\left (c-a^2 c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6749

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

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

input

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

output

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

3.8.55.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 88

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)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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]
 

3.8.55.4 Maple [F]

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

input

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

output

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

3.8.55.5 Fricas [F]

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

input

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

output

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

3.8.55.6 Sympy [F]

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

input

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

output

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

3.8.55.7 Maxima [F]

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

input

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

output

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

3.8.55.8 Giac [F]

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

input

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

output

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

3.8.55.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

input

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

output

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