Integrand size = 25, antiderivative size = 123 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac {2 a (1-a x)^{-n/2} (1+a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {1+a x}{1-a x}\right )}{c} \] Output:
a*(1+n)*(a*x+1)^(1/2*n)/c/n/((-a*x+1)^(1/2*n))-(a*x+1)^(1/2*n)/c/x/((-a*x+ 1)^(1/2*n))-2*a*(a*x+1)^(1/2*n)*hypergeom([1, 1/2*n],[1+1/2*n],(a*x+1)/(-a *x+1))/c/((-a*x+1)^(1/2*n))
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {(1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}} \left ((-2+n) (1+a x) (a x+n (-1+a x))-2 a n^2 x (-1+a x) \operatorname {Hypergeometric2F1}\left (1,1-\frac {n}{2},2-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{c (-2+n) n x} \] Input:
Integrate[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)),x]
Output:
((1 + a*x)^(-1 + n/2)*((-2 + n)*(1 + a*x)*(a*x + n*(-1 + a*x)) - 2*a*n^2*x *(-1 + a*x)*Hypergeometric2F1[1, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)]))/ (c*(-2 + n)*n*x*(1 - a*x)^(n/2))
Time = 0.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6700, 144, 25, 27, 172, 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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\int \frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}}{x^2}dx}{c}\) |
\(\Big \downarrow \) 144 |
\(\displaystyle \frac {-\int -\frac {a (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} (n+a x)}{x}dx-\frac {(a x+1)^{n/2} (1-a x)^{-n/2}}{x}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} (n+a x)}{x}dx-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} (n+a x)}{x}dx-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
\(\Big \downarrow \) 172 |
\(\displaystyle \frac {a \left (\frac {(n+1) (1-a x)^{-n/2} (a x+1)^{n/2}}{n}-\frac {\int -\frac {a n^2 (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{x}dx}{a n}\right )-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {a \left (\frac {\int \frac {a n^2 (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{x}dx}{a n}+\frac {(n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{n}\right )-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \left (n \int \frac {(1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}}{x}dx+\frac {(n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{n}\right )-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle \frac {a \left (\frac {(n+1) (1-a x)^{-n/2} (a x+1)^{n/2}}{n}-2 (1-a x)^{-n/2} (a x+1)^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a x+1}{1-a x}\right )\right )-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{x}}{c}\) |
Input:
Int[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)),x]
Output:
(-((1 + a*x)^(n/2)/(x*(1 - a*x)^(n/2))) + a*(((1 + n)*(1 + a*x)^(n/2))/(n* (1 - a*x)^(n/2)) - (2*(1 + a*x)^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - a*x)^(n/2)))/c
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + 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*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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
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])
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )}d x\]
Input:
int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x)
Output:
int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x^{2}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x, algorithm="fricas")
Output:
integral(-(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^4 - c*x^2), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=- \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{4} - x^{2}}\, dx}{c} \] Input:
integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c),x)
Output:
-Integral(exp(n*atanh(a*x))/(a**2*x**4 - x**2), x)/c
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x^{2}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x, algorithm="maxima")
Output:
-integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/((a^2*c*x^2 - c)*x^2), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x^{2}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c),x, algorithm="giac")
Output:
integrate(-(-(a*x + 1)/(a*x - 1))^(1/2*n)/((a^2*c*x^2 - c)*x^2), x)
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,\left (c-a^2\,c\,x^2\right )} \,d x \] Input:
int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)),x)
Output:
int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} a x -e^{\mathit {atanh} \left (a x \right ) n} n -\left (\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{a^{2} x^{3}-x}d x \right ) a \,n^{2} x}{c n x} \] Input:
int(exp(n*atanh(a*x))/x^2/(-a^2*c*x^2+c),x)
Output:
(e**(atanh(a*x)*n)*a*x - e**(atanh(a*x)*n)*n - int(e**(atanh(a*x)*n)/(a**2 *x**3 - x),x)*a*n**2*x)/(c*n*x)