\(\int e^{n \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [763]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {4 c (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-a x}{1+a x}\right )}{a (2-n)}-\frac {2^{1+\frac {n}{2}} c (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c e^{n \text {arctanh}(a x)} \left (2+n+a e^{2 \text {arctanh}(a x)} n x \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )+a (2+n) x \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \text {arctanh}(a x)}\right )+4 a e^{2 \text {arctanh}(a x)} x \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \text {arctanh}(a x)}\right )\right )}{a^2 (2+n) x} \] Input:

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

Output:

(c*E^(n*ArcTanh[a*x])*(2 + n + a*E^(2*ArcTanh[a*x])*n*x*Hypergeometric2F1[ 
1, 1 + n/2, 2 + n/2, E^(2*ArcTanh[a*x])] + a*(2 + n)*x*Hypergeometric2F1[1 
, n/2, 1 + n/2, E^(2*ArcTanh[a*x])] + 4*a*E^(2*ArcTanh[a*x])*x*Hypergeomet 
ric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcTanh[a*x])]))/(a^2*(2 + n)*x)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.64, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6707, 6700, 139, 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.

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

\(\Big \downarrow \) 6707

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 139

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

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 6700

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

rule 6707

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(exp(n*arctanh(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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(exp(n*arctanh(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)
 

Giac [F]

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

integrate(exp(n*arctanh(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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(c*(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))/(a**2*n)