\(\int e^{n \text {arctanh}(a x)} (c-\frac {c}{a x})^2 \, dx\) [592]

Optimal result

Integrand size = 22, antiderivative size = 130 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4 c^2 (1-a x)^{-n/2} (1+a x)^{n/2} \operatorname {Hypergeometric2F1}\left (2,\frac {n}{2},\frac {2+n}{2},\frac {1+a x}{1-a x}\right )}{a n}+\frac {2^{n/2} c^2 (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (4-n)} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(262\) vs. \(2(130)=260\).

Time = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.02 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 e^{n \text {arctanh}(a x)} \left (2 n+n^2-2 a e^{2 \text {arctanh}(a x)} n x \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )+a e^{2 \text {arctanh}(a x)} (-2+n) n x \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )+4 a x \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )+2 a n x \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )-4 a x \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )+a n^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )-4 a e^{2 \text {arctanh}(a x)} n x \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )\right )}{a^2 n (2+n) x} \] Input:

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

Output:

-((c^2*E^(n*ArcTanh[a*x])*(2*n + n^2 - 2*a*E^(2*ArcTanh[a*x])*n*x*Hypergeo 
metric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])] + a*E^(2*ArcTanh[a*x]) 
*(-2 + n)*n*x*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcTanh[a*x])] + 
 4*a*x*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcTanh[a*x])] + 2*a*n*x*H 
ypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcTanh[a*x])] - 4*a*x*Hypergeomet 
ric2F1[1, n/2, 1 + n/2, E^(2*ArcTanh[a*x])] + a*n^2*x*Hypergeometric2F1[1, 
 n/2, 1 + n/2, E^(2*ArcTanh[a*x])] - 4*a*E^(2*ArcTanh[a*x])*n*x*Hypergeome 
tric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])]))/(a^2*n*(2 + n)*x))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6681, 6679, 138, 79, 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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 138

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

(c^2*((4*a*(1 + a*x)^(n/2)*Hypergeometric2F1[2, n/2, (2 + n)/2, (1 + a*x)/ 
(1 - a*x)])/(n*(1 - a*x)^(n/2)) + (2^(n/2)*a*(1 - a*x)^(2 - n/2)*Hypergeom 
etric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (1 - a*x)/2])/(4 - n)))/a^2
 

Defintions of rubi rules used

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 138

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

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 6679

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

rule 6681

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (\int a^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\, dx + \int \left (- \frac {2 a e^{n \operatorname {atanh}{\left (a x \right )}}}{x}\right )\, dx\right )}{a^{2}} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (-e^{\mathit {atanh} \left (a x \right ) n} a +\left (\int e^{\mathit {atanh} \left (a x \right ) n}d x \right ) a^{2} n -\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{2} x^{4}-x^{2}}d x \right ) n -2 \left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{x}d x \right ) a n \right )}{a^{2} n} \] Input:

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

Output:

(c**2*( - e**(atanh(a*x)*n)*a + int(e**(atanh(a*x)*n),x)*a**2*n - int(e**( 
atanh(a*x)*n)/(a**2*x**4 - x**2),x)*n - 2*int(e**(atanh(a*x)*n)/x,x)*a*n)) 
/(a**2*n)