3.10.28 \(\int e^{n \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [928]

3.10.28.1 Optimal result

Integrand size = 20, antiderivative size = 154 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {4 c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (2-n)}-\frac {2^{1+\frac {n}{2}} c \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{a (2-n)} \]

output

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

3.10.28.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.80 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c e^{n \coth ^{-1}(a x)} \left (2 a x+a n x+e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+4 e^{2 \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )}{a (2+n)} \]

input

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

output

(c*E^(n*ArcCoth[a*x])*(2*a*x + a*n*x + E^(2*ArcCoth[a*x])*n*Hypergeometric 
2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*Hypergeometric2F1[1 
, n/2, 1 + n/2, E^(2*ArcCoth[a*x])] + 4*E^(2*ArcCoth[a*x])*Hypergeometric2 
F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]))/(a*(2 + n))
 

3.10.28.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.68, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6748, 139, 27, 88, 79, 168, 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.

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

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 139

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

\(\displaystyle -c \left (\frac {\int \left (a+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} x^2d\frac {1}{x}}{a}+\frac {\frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(2-n) (4-n)}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{a^3}\right )\)

\(\Big \downarrow \) 168

\(\displaystyle -c \left (\frac {-\int -n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} xd\frac {1}{x}-a x \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}}}{a}+\frac {\frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(2-n) (4-n)}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{a^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -c \left (\frac {\int n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} xd\frac {1}{x}-a x \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{a}+\frac {\frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(2-n) (4-n)}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{a^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -c \left (\frac {n \int \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} xd\frac {1}{x}-a x \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{a}+\frac {\frac {a^2 2^{n/2} n \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{(2-n) (4-n)}-\frac {2 a^2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{a^3}\right )\)

\(\Big \downarrow \) 141

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

input

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

output

-(c*((-(a*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*x) + (2*n*(1 
- 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((-2 + n)/2)*Hypergeometric2F1[1, (-2 
 + n)/2, n/2, (a + x^(-1))/(a - x^(-1))])/(2 - n))/a + ((-2*a^2*(1 - 1/(a* 
x))^(2 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(2 - n) + (2^(n/2)*a^2*n*(1 - 1/ 
(a*x))^(2 - n/2)*Hypergeometric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (a - x^(-1 
))/(2*a)])/((2 - n)*(4 - n)))/a^3))
 

3.10.28.3.1 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 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 139

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

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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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]
 

3.10.28.4 Maple [F]

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

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)
 

3.10.28.5 Fricas [F]

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

input

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

output

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

3.10.28.6 Sympy [F]

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

input

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

output

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

3.10.28.7 Maxima [F]

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

input

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

output

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

3.10.28.8 Giac [F]

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

input

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

output

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

3.10.28.9 Mupad [F(-1)]

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

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)