Integrand size = 27, antiderivative size = 243 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c (1+n) \sqrt {c-a^2 c x^2}}-\frac {(2+n) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2}}{c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {2 (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {1+a x}{1-a x}\right )}{c (1-n) \sqrt {c-a^2 c x^2}} \]
(-a*x+1)^(-1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c/(1+n)/(-a^ 2*c*x^2+c)^(1/2)-(2+n)*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-1/2+1/2*n)*(-a^2*x^2 +1)^(1/2)/c/(-n^2+1)/(-a^2*c*x^2+c)^(1/2)+2*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^( -1/2+1/2*n)*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a*x+1)/(-a*x+1))*(-a^2* x^2+1)^(1/2)/c/(1-n)/(-a^2*c*x^2+c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.61 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (-((-3+n) (1+a x) (-1+2 a x+n (-2+a x)))+2 \left (-1+n^2\right ) (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{c (-3+n) (-1+n) (1+n) \sqrt {c-a^2 c x^2}} \]
((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(-((-3 + n)*(1 + a*x)*(-1 + 2*a*x + n*(-2 + a*x))) + 2*(-1 + n^2)*(-1 + a*x)^2*Hype rgeometric2F1[1, 3/2 - n/2, 5/2 - n/2, (1 - a*x)/(1 + a*x)]))/(c*(-3 + n)* (-1 + n)*(1 + n)*Sqrt[c - a^2*c*x^2])
Time = 0.57 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6703, 6700, 144, 25, 27, 172, 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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}}dx}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}dx}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 144 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}}{n+1}-\frac {\int -\frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}} (n+a x+1)}{x}dx}{a (n+1)}\right )}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\int \frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}} (n+a x+1)}{x}dx}{a (n+1)}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}\right )}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\int \frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}} (n+a x+1)}{x}dx}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}\right )}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 172 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\frac {\int \frac {a \left (1-n^2\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx}{a (1-n)}-\frac {(n+2) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{1-n}}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}\right )}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\frac {\left (1-n^2\right ) \int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx}{1-n}-\frac {(n+2) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{1-n}}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}\right )}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {\frac {2 \left (1-n^2\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a x+1}{1-a x}\right )}{(1-n)^2}-\frac {(n+2) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-1}{2}}}{1-n}}{n+1}+\frac {(a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}\right )}{c \sqrt {c-a^2 c x^2}}\) |
(Sqrt[1 - a^2*x^2]*(((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(1 + n ) + (-(((2 + n)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2))/(1 - n)) + ( 2*(1 - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Hypergeometric2F1 [1, (-1 + n)/2, (1 + n)/2, (1 + a*x)/(1 - a*x)])/(1 - n)^2)/(1 + n)))/(c*S qrt[c - a^2*c*x^2])
3.14.45.3.1 Defintions of rubi rules used
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[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]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
integral(sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x^5 - 2*a^2*c^2*x^3 + c^2*x), x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x} \,d x } \]
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]