3.10.29 \(\int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [929]

3.10.29.1 Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {(1+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2}}{a c n}+\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} x}{c}+\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c} \]

output

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

3.10.29.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.63 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n^2 \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a n x+n \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a c n (2+n)} \]

input

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

output

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

3.10.29.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6748, 144, 25, 27, 172, 25, 27, 141}

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.

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

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

input

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

output

-((-(((1 + 1/(a*x))^(n/2)*x)/(1 - 1/(a*x))^(n/2)) + ((a*(1 + n)*(1 + 1/(a* 
x))^(n/2))/(n*(1 - 1/(a*x))^(n/2)) - (2*a*(1 + 1/(a*x))^(n/2)*Hypergeometr 
ic2F1[1, n/2, (2 + n)/2, (a + x^(-1))/(a - x^(-1))])/(1 - 1/(a*x))^(n/2))/ 
a^2)/c)
 

3.10.29.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 141

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

rule 144

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

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) 
*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

rule 6748

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> 
 Simp[-c^p   Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^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]
 

3.10.29.4 Maple [F]

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

input

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

output

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

3.10.29.5 Fricas [F]

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

input

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

output

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

3.10.29.6 Sympy [F]

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

input

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

output

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

3.10.29.7 Maxima [F]

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

input

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

output

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

3.10.29.8 Giac [F]

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

input

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

output

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

3.10.29.9 Mupad [F(-1)]

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

input

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

output

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