\(\int e^{n \text {arctanh}(a x)} x^3 \, dx\) [149]

Optimal result

Integrand size = 12, antiderivative size = 187 \[ \int e^{n \text {arctanh}(a x)} x^3 \, dx=-\frac {(6+n (2+n)) (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{24 a^4}-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{4 a^2}+\frac {n (1-a x)^{2-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{12 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{3 a^4 (2-n)} \] Output:

-1/24*(6+n*(2+n))*(-a*x+1)^(1-1/2*n)*(a*x+1)^(1+1/2*n)/a^4-1/4*x^2*(-a*x+1 
)^(1-1/2*n)*(a*x+1)^(1+1/2*n)/a^2+1/12*n*(-a*x+1)^(2-1/2*n)*(a*x+1)^(1+1/2 
*n)/a^4-1/3*2^(-2+1/2*n)*n*(n^2+8)*(-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^4/(2-n)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97 \[ \int e^{n \text {arctanh}(a x)} x^3 \, dx=-\frac {(1-a x)^{1-\frac {n}{2}} \left (-2^{3+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (-2-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )+2^{3+\frac {n}{2}} (-1+n) \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )+(-2+n) \left (a^2 x^2 (1+a x)^{1+\frac {n}{2}}-2^{1+\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )\right )}{4 a^4 (-2+n)} \] Input:

Integrate[E^(n*ArcTanh[a*x])*x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6676, 111, 25, 164, 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.

\(\displaystyle \int x^3 e^{n \text {arctanh}(a x)} \, dx\)

\(\Big \downarrow \) 6676

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {n \left (n^2+8\right ) \int (1-a x)^{-n/2} (a x+1)^{n/2}dx}{6 a}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}} \left (2 a n x+n^2+6\right )}{6 a^2}}{4 a^2}-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}}}{4 a^2}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {-\frac {2^{n/2} n \left (n^2+8\right ) (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{3 a^2 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} \left (2 a n x+n^2+6\right ) (1-a x)^{1-\frac {n}{2}}}{6 a^2}}{4 a^2}-\frac {x^2 (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}}}{4 a^2}\)

Input:

Int[E^(n*ArcTanh[a*x])*x^3,x]
 

Output:

-1/4*(x^2*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/a^2 + (-1/6*((1 - a*x 
)^(1 - n/2)*(1 + a*x)^((2 + n)/2)*(6 + n^2 + 2*a*n*x))/a^2 - (2^(n/2)*n*(8 
 + n^2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 
 - a*x)/2])/(3*a^2*(2 - n)))/(4*a^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 164

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

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{3}d x\]

Input:

int(exp(n*arctanh(a*x))*x^3,x)
 

Output:

int(exp(n*arctanh(a*x))*x^3,x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))*x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} x^3 \, dx=\int x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:

integrate(exp(n*atanh(a*x))*x**3,x)
 

Output:

Integral(x**3*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))*x^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(n*arctanh(a*x))*x^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^3*exp(n*atanh(a*x)),x)
 

Output:

int(x^3*exp(n*atanh(a*x)), x)
 

Reduce [F]

\[ \int e^{n \text {arctanh}(a x)} x^3 \, dx=\int e^{\mathit {atanh} \left (a x \right ) n} x^{3}d x \] Input:

int(exp(n*atanh(a*x))*x^3,x)
 

Output:

int(e**(atanh(a*x)*n)*x**3,x)