Integrand size = 12, antiderivative size = 150 \[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=-\frac {3 x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m}+\frac {4 x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}-\frac {4 a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m} \]
-3*x^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)+a*x^(2+m) *hypergeom([1/2, 1+1/2*m],[2+1/2*m],a^2*x^2)/(2+m)+4*x^(1+m)*hypergeom([3/ 2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)-4*a*x^(2+m)*hypergeom([3/2, 1+1/2 *m],[2+1/2*m],a^2*x^2)/(2+m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.37 \[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=-\frac {x^{1+m} \left (\operatorname {AppellF1}\left (1+m,-\frac {1}{2},\frac {1}{2},2+m,a x,-a x\right )-2 \operatorname {AppellF1}\left (1+m,-\frac {1}{2},\frac {3}{2},2+m,a x,-a x\right )\right )}{1+m} \]
-((x^(1 + m)*(AppellF1[1 + m, -1/2, 1/2, 2 + m, a*x, -(a*x)] - 2*AppellF1[ 1 + m, -1/2, 3/2, 2 + m, a*x, -(a*x)]))/(1 + m))
Time = 0.56 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6674, 2355, 557, 278, 583, 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.
\(\displaystyle \int x^m e^{-3 \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {(1-a x)^2 x^m}{(a x+1) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 2355 |
\(\displaystyle \int \frac {x^m (a x-3)}{\sqrt {1-a^2 x^2}}dx+4 \int \frac {x^m}{(a x+1) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 557 |
\(\displaystyle a \int \frac {x^{m+1}}{\sqrt {1-a^2 x^2}}dx-3 \int \frac {x^m}{\sqrt {1-a^2 x^2}}dx+4 \int \frac {x^m}{(a x+1) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle 4 \int \frac {x^m}{(a x+1) \sqrt {1-a^2 x^2}}dx-\frac {3 x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}\) |
\(\Big \downarrow \) 583 |
\(\displaystyle 4 \int \frac {x^m (1-a x)}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {3 x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}\) |
\(\Big \downarrow \) 557 |
\(\displaystyle 4 \left (\int \frac {x^m}{\left (1-a^2 x^2\right )^{3/2}}dx-a \int \frac {x^{m+1}}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {3 x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {3 x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}+4 \left (\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}-\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}\right )\) |
(-3*x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/( 2 + m) + 4*((x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, a^2*x^ 2])/(1 + m) - (a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, a^ 2*x^2])/(2 + m))
3.2.47.3.1 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_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, 0]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] I nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} , x] && PolynomialQ[Px, x] && LtQ[n, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x )^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / ; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
\[\int \frac {x^{m} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a x +1\right )^{3}}d x\]
\[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{{\left (a x + 1\right )}^{3}} \,d x } \]
\[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=\int \frac {x^{m} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
\[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{{\left (a x + 1\right )}^{3}} \,d x } \]
Exception generated. \[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{-3 \text {arctanh}(a x)} x^m \, dx=\int \frac {x^m\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]