\(\int \frac {e^{n \text {arctanh}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx\) [1372]

Optimal result

Integrand size = 27, antiderivative size = 322 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{a^4 (1-n) \sqrt {c-a^2 c x^2}}+\frac {n (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{a^4 \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {(1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {3+n}{2}} \sqrt {1-a^2 x^2}}{3 a^4 \sqrt {c-a^2 c x^2}}+\frac {2^{\frac {3}{2}-\frac {n}{2}} n \left (5+n^2\right ) (1+a x)^{\frac {3+n}{2}} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1+n),\frac {3+n}{2},\frac {5+n}{2},\frac {1}{2} (1+a x)\right )}{3 a^4 (1-n) (1+n) (3+n) \sqrt {c-a^2 c x^2}} \] Output:

-(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(1/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a^4/(1-n)/(-a 
^2*c*x^2+c)^(1/2)+n*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(1/2+1/2*n)*(-a^2*x^2+1)^ 
(1/2)/a^4/(-n^2+1)/(-a^2*c*x^2+c)^(1/2)+1/3*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^( 
3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a^4/(-a^2*c*x^2+c)^(1/2)+1/3*2^(3/2-1/2*n)*n 
*(n^2+5)*(a*x+1)^(3/2+1/2*n)*(-a^2*x^2+1)^(1/2)*hypergeom([-1/2+1/2*n, 3/2 
+1/2*n],[5/2+1/2*n],1/2*a*x+1/2)/a^4/(1-n)/(1+n)/(3+n)/(-a^2*c*x^2+c)^(1/2 
)
 

Mathematica [A] (warning: unable to verify)

Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.58 \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx=\frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-a^2 x^2} \left (-\sqrt {2} (-3+n) (1+a x)^{\frac {1+n}{2}} \left (n^2 (-1+a x)-2 \left (2+a^2 x^2\right )+n \left (-1-a x+2 a^2 x^2\right )\right )+2^{1+\frac {n}{2}} n \left (5+n^2\right ) (-1+a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )\right )}{6 \sqrt {2} a^4 (-3+n) (-1+n) \sqrt {c-a^2 c x^2}} \] 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)*Sqrt[1 - a^2*x^2]*(-(Sqrt[2]*(-3 + n)*(1 + a*x)^((1 
 + n)/2)*(n^2*(-1 + a*x) - 2*(2 + a^2*x^2) + n*(-1 - a*x + 2*a^2*x^2))) + 
2^(1 + n/2)*n*(5 + n^2)*(-1 + a*x)*Hypergeometric2F1[1/2 - n/2, 3/2 - n/2, 
 5/2 - n/2, 1/2 - (a*x)/2]))/(6*Sqrt[2]*a^4*(-3 + n)*(-1 + n)*Sqrt[c - a^2 
*c*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.70, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6703, 6700, 111, 25, 163, 79}

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.

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 163

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\frac {n \left (n^2+5\right ) \int (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}dx}{2 a (1-n)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n+1}{2}} \left (a (1-n) n x+n^2+n+4\right )}{2 a^2 (1-n)}}{3 a^2}-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n+1}{2}}}{3 a^2}\right )}{\sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {-\frac {2^{\frac {n+1}{2}-1} n \left (n^2+5\right ) (1-a x)^{\frac {3-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {3-n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (1-n) (3-n)}-\frac {(a x+1)^{\frac {n+1}{2}} \left (a (1-n) n x+n^2+n+4\right ) (1-a x)^{\frac {1-n}{2}}}{2 a^2 (1-n)}}{3 a^2}-\frac {x^2 (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n+1}{2}}}{3 a^2}\right )}{\sqrt {c-a^2 c x^2}}\)

Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(-1/3*(x^2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)) 
/a^2 + (-1/2*((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*(4 + n + n^2 + a 
*(1 - n)*n*x))/(a^2*(1 - n)) - (2^(-1 + (1 + n)/2)*n*(5 + n^2)*(1 - a*x)^( 
(3 - n)/2)*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2] 
)/(a^2*(1 - n)*(3 - n)))/(3*a^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 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 111

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

rule 163

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n 
+ 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* 
(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f 
*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* 
d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* 
d*(b*c - a*d)*(m + 1)*(m + n + 3))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 
1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
 

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]

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \text {arctanh}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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