Integrand size = 23, antiderivative size = 80 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{c^2 (1+m)}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{c^2 (2+m)} \]
x^(1+m)*hypergeom([5/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/c^2/(1+m)+a*x^(2+m )*hypergeom([5/2, 1+1/2*m],[2+1/2*m],a^2*x^2)/c^2/(2+m)
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},1+\frac {1+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {2+m}{2},1+\frac {2+m}{2},a^2 x^2\right )}{2+m}}{c^2} \]
((x^(1 + m)*Hypergeometric2F1[5/2, (1 + m)/2, 1 + (1 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, 1 + (2 + m)/2, a^2*x ^2])/(2 + m))/c^2
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6698, 557, 278}
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.
\(\displaystyle \int \frac {x^m e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^m (a x+1)}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
\(\Big \downarrow \) 557 |
\(\displaystyle \frac {a \int \frac {x^{m+1}}{\left (1-a^2 x^2\right )^{5/2}}dx+\int \frac {x^m}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}}{c^2}\) |
((x^(1 + m)*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m))/c^2
3.10.90.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
\[\int \frac {\left (a x +1\right ) x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )^{2}}d x\]
\[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}} \,d x } \]
integral(-sqrt(-a^2*x^2 + 1)*x^m/(a^5*c^2*x^5 - a^4*c^2*x^4 - 2*a^3*c^2*x^ 3 + 2*a^2*c^2*x^2 + a*c^2*x - c^2), x)
\[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{m}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x x^{m}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
(Integral(x**m/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x* *2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x*x**m/(a**4*x**4*sqrt(-a **2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2
\[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}} \,d x } \]
\[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)} x^m}{\left (c-a^2 c x^2\right )^2} \, dx=\int \frac {x^m\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^2\,\sqrt {1-a^2\,x^2}} \,d x \]