Integrand size = 10, antiderivative size = 99 \[ \int e^{n \text {arctanh}(a x)} x \, dx=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 a^2}-\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (2-n)} \] Output:
-1/2*(-a*x+1)^(1-1/2*n)*(a*x+1)^(1+1/2*n)/a^2-2^(1/2*n)*n*(-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/(2-n)
Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int e^{n \text {arctanh}(a x)} x \, dx=-\frac {(1-a x)^{1-\frac {n}{2}} \left ((-2+n) (1+a x)^{1+\frac {n}{2}}-2^{1+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{2 a^2 (-2+n)} \] Input:
Integrate[E^(n*ArcTanh[a*x])*x,x]
Output:
-1/2*((1 - a*x)^(1 - n/2)*((-2 + n)*(1 + a*x)^(1 + n/2) - 2^(1 + n/2)*n*Hy pergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - a*x)/2]))/(a^2*(-2 + n))
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6676, 90, 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.
\(\displaystyle \int x e^{n \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6676 |
\(\displaystyle \int x (1-a x)^{-n/2} (a x+1)^{n/2}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {n \int (1-a x)^{-n/2} (a x+1)^{n/2}dx}{2 a}-\frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}}}{2 a^2}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{n/2} n (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (2-n)}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 a^2}\) |
Input:
Int[E^(n*ArcTanh[a*x])*x,x]
Output:
-1/2*((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/a^2 - (2^(n/2)*n*(1 - a*x )^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - a*x)/2])/(a^2 *(2 - n))
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]))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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]
\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x d x\]
Input:
int(exp(n*arctanh(a*x))*x,x)
Output:
int(exp(n*arctanh(a*x))*x,x)
\[ \int e^{n \text {arctanh}(a x)} x \, dx=\int { x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x,x, algorithm="fricas")
Output:
integral(x*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \text {arctanh}(a x)} x \, dx=\int x e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:
integrate(exp(n*atanh(a*x))*x,x)
Output:
Integral(x*exp(n*atanh(a*x)), x)
\[ \int e^{n \text {arctanh}(a x)} x \, dx=\int { x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x,x, algorithm="maxima")
Output:
integrate(x*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \text {arctanh}(a x)} x \, dx=\int { x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*x,x, algorithm="giac")
Output:
integrate(x*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
Timed out. \[ \int e^{n \text {arctanh}(a x)} x \, dx=\int x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )} \,d x \] Input:
int(x*exp(n*atanh(a*x)),x)
Output:
int(x*exp(n*atanh(a*x)), x)
\[ \int e^{n \text {arctanh}(a x)} x \, dx=\int e^{\mathit {atanh} \left (a x \right ) n} x d x \] Input:
int(exp(n*atanh(a*x))*x,x)
Output:
int(e**(atanh(a*x)*n)*x,x)