\(\int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx\) [1378]

Optimal result

Integrand size = 27, antiderivative size = 242 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{2 x \sqrt {c-a^2 c x^2}}-\frac {a^2 \left (1+n^2\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {1-a x}{1+a x}\right )}{(1-n) \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.55 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \left (-((-1+n) (1+a x) (1+a n x))+2 a^2 \left (1+n^2\right ) x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{2 (-1+n) x^2 \sqrt {c-a^2 c x^2}} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(x^3*Sqrt[c - a^2*c*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6703, 6700, 144, 25, 27, 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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {1-a^2 x^2}}dx}{\sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

Input:

Int[E^(n*ArcTanh[a*x])/(x^3*Sqrt[c - a^2*c*x^2]),x]
 

Output:

(Sqrt[1 - a^2*x^2]*(-1/2*((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2))/x^2 
 + (a*(-((n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2))/x) - (2*a*(1 + n^ 
2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Hypergeometric2F1[1, (1 - 
n)/2, (3 - n)/2, (1 - a*x)/(1 + a*x)])/(1 - n)))/2))/Sqrt[c - a^2*c*x^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 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 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 6703

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{x^{3} \sqrt {-a^{2} c \,x^{2}+c}}d x\]

Input:

int(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x)
 

Output:

int(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c} x^{3}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm="frica 
s")
 

Output:

integral(-sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^5 - 
 c*x^3), x)
 

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(exp(n*atanh(a*x))/x**3/(-a**2*c*x**2+c)**(1/2),x)
 

Output:

Integral(exp(n*atanh(a*x))/(x**3*sqrt(-c*(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c} x^{3}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm="maxim 
a")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(sqrt(-a^2*c*x^2 + c)*x^3), x)
 

Giac [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c} x^{3}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(sqrt(-a^2*c*x^2 + c)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^3\,\sqrt {c-a^2\,c\,x^2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(x^3*(c - a^2*c*x^2)^(1/2)),x)
 

Output:

int(exp(n*atanh(a*x))/(x^3*(c - a^2*c*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x}{\sqrt {c}} \] Input:

int(exp(n*atanh(a*x))/x^3/(-a^2*c*x^2+c)^(1/2),x)
 

Output:

int(e**(atanh(a*x)*n)/(sqrt( - a**2*x**2 + 1)*x**3),x)/sqrt(c)