\(\int e^{n \text {arctanh}(a x)} (c-\frac {c}{a x}) \, dx\) [608]

Optimal result

Integrand size = 20, antiderivative size = 187 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c (1-a x)^{2-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a (2-n)}-\frac {2 c (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-2+n),\frac {n}{2},\frac {1+a x}{1-a x}\right )}{a (2-n)}+\frac {2^{n/2} c (1-n) (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n) (4-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.80 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {2 c (1-a x)^{-n/2} \left ((-2+n) (1+a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-a x}{1+a x}\right )+2^{n/2} \left (n (-1+a x) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )-(-2+n) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )\right )}{a (-2+n) n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6681, 6679, 139, 88, 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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 139

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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