Integrand size = 25, antiderivative size = 176 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {(1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1+n}{2}} \sqrt {1-a^2 x^2}}{a^2 (1+n) \sqrt {c-a^2 c x^2}}-\frac {2^{\frac {3+n}{2}} n (1-a x)^{\frac {1-n}{2}} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-n),\frac {1-n}{2},\frac {3-n}{2},\frac {1}{2} (1-a x)\right )}{a^2 \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
-(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(1/2+1/2*n)*(-a^2*x^2+1)^(1/2)/a^2/(1+n)/(-a ^2*c*x^2+c)^(1/2)-2^(3/2+1/2*n)*n*(-a*x+1)^(1/2-1/2*n)*hypergeom([1/2-1/2* n, -1/2-1/2*n],[3/2-1/2*n],-1/2*a*x+1/2)*(-a^2*x^2+1)^(1/2)/a^2/(-n^2+1)/( -a^2*c*x^2+c)^(1/2)
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} \sqrt {1-a^2 x^2} \left (-\left ((-1+n) (1+a x)^{\frac {1+n}{2}}\right )+2^{\frac {3+n}{2}} n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-\frac {n}{2},\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )\right )}{a^2 \left (-1+n^2\right ) \sqrt {c-a^2 c x^2}} \]
((1 - a*x)^(1/2 - n/2)*Sqrt[1 - a^2*x^2]*(-((-1 + n)*(1 + a*x)^((1 + n)/2) ) + 2^((3 + n)/2)*n*Hypergeometric2F1[-1/2 - n/2, 1/2 - n/2, 3/2 - n/2, 1/ 2 - (a*x)/2]))/(a^2*(-1 + n^2)*Sqrt[c - a^2*c*x^2])
Time = 0.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6703, 6700, 88, 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.
\(\displaystyle \int \frac {x e^{n \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {1-a^2 x^2}}dx}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int x (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-1}{2}}dx}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {n \int (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n+1}{2}}dx}{a (n+1)}-\frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n+1}{2}}}{a^2 (n+1)}\right )}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (-\frac {2^{\frac {n+3}{2}} n (1-a x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {1-n}{2},\frac {3-n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (1-n) (n+1)}-\frac {(a x+1)^{\frac {n+1}{2}} (1-a x)^{\frac {1-n}{2}}}{a^2 (n+1)}\right )}{\sqrt {c-a^2 c x^2}}\) |
(Sqrt[1 - a^2*x^2]*(-(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2))/(a^2*( 1 + n))) - (2^((3 + n)/2)*n*(1 - a*x)^((1 - n)/2)*Hypergeometric2F1[(-1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/(a^2*(1 - n)*(1 + n))))/Sqrt[c - a^2*c*x^2]
3.14.36.3.1 Defintions of rubi rules used
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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 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]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x}{\sqrt {-a^{2} c \,x^{2}+c}}d x\]
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {x e^{n \operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{\sqrt {c-a^2\,c\,x^2}} \,d x \]