Integrand size = 12, antiderivative size = 109 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac {\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}} x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-m),-\frac {m}{2},\frac {a^2}{x^2}\right )}{a \left (2+3 m+m^2\right )} \]
(x/a+(-1+x/a)^(1/2)*(1+x/a)^(1/2))*x^(1+m)/(1+m)-x^(2+m)/a/(m^2+3*m+2)-x^( 2+m)*hypergeom([1/2, -1-1/2*m],[-1/2*m],a^2/x^2)*(1/(1+a/x))^(1/2)*(1+a/x) ^(1/2)/a/(m^2+3*m+2)
Time = 0.71 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=-\frac {2^{-1-m} a e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} \left (\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{1+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}}\right )^{-1-m} \left (\frac {a}{x}\right )^m x^m \left (-\left ((-2+m) \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},1-\frac {m}{2},-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )} m \operatorname {Hypergeometric2F1}\left (1,2+\frac {m}{2},2-\frac {m}{2},-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )}{(-2+m) m} \]
-((2^(-1 - m)*a*E^ArcSech[a/x]*(E^ArcSech[a/x]/(1 + E^(2*ArcSech[a/x])))^( -1 - m)*(a/x)^m*x^m*(-((-2 + m)*Hypergeometric2F1[1, 1 + m/2, 1 - m/2, -E^ (2*ArcSech[a/x])]) + E^(2*ArcSech[a/x])*m*Hypergeometric2F1[1, 2 + m/2, 2 - m/2, -E^(2*ArcSech[a/x])]))/((-2 + m)*m))
Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6889, 15, 791, 862, 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.
\(\displaystyle \int x^m e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle -\frac {\int x^{m+1}dx}{a (m+1)}-\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} \int \frac {x^{m+1}}{\sqrt {1-\frac {a}{x}} \sqrt {\frac {a}{x}+1}}dx}{a (m+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} \int \frac {x^{m+1}}{\sqrt {1-\frac {a}{x}} \sqrt {\frac {a}{x}+1}}dx}{a (m+1)}-\frac {x^{m+2}}{a (m+1) (m+2)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\) |
\(\Big \downarrow \) 791 |
\(\displaystyle -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} \int \frac {x^{m+1}}{\sqrt {1-\frac {a^2}{x^2}}}dx}{a (m+1)}-\frac {x^{m+2}}{a (m+1) (m+2)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\) |
\(\Big \downarrow \) 862 |
\(\displaystyle \frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} \left (\frac {1}{x}\right )^m x^m \int \frac {\left (\frac {1}{x}\right )^{-m-3}}{\sqrt {1-\frac {a^2}{x^2}}}d\frac {1}{x}}{a (m+1)}-\frac {x^{m+2}}{a (m+1) (m+2)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-2),-\frac {m}{2},\frac {a^2}{x^2}\right )}{a (m+1) (m+2)}-\frac {x^{m+2}}{a (m+1) (m+2)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\) |
(E^ArcSech[a/x]*x^(1 + m))/(1 + m) - x^(2 + m)/(a*(1 + m)*(2 + m)) - (Sqrt [(1 + a/x)^(-1)]*Sqrt[1 + a/x]*x^(2 + m)*Hypergeometric2F1[1/2, (-2 - m)/2 , -1/2*m, a^2/x^2])/(a*(1 + m)*(2 + m))
3.1.59.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) ^(n_))^(p_), x_Symbol] :> Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; Free Q[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ (-1))*(c*x)^(m + 1)*(1/x)^(m + 1) Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && !RationalQ[m]
Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ ArcSech[a*x^p]/(m + 1)), x] + (Simp[p/(a*(m + 1)) Int[x^(m - p), x], x] + Simp[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)] Int[x^(m - p)/( Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
\[\int \left (\frac {x}{a}+\sqrt {-1+\frac {x}{a}}\, \sqrt {1+\frac {x}{a}}\right ) x^{m}d x\]
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {x}{a} + 1} \sqrt {\frac {x}{a} - 1} + \frac {x}{a}\right )} \,d x } \]
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\frac {\int x x^{m}\, dx + \int a x^{m} \sqrt {-1 + \frac {x}{a}} \sqrt {1 + \frac {x}{a}}\, dx}{a} \]
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {x}{a} + 1} \sqrt {\frac {x}{a} - 1} + \frac {x}{a}\right )} \,d x } \]
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {x}{a} + 1} \sqrt {\frac {x}{a} - 1} + \frac {x}{a}\right )} \,d x } \]
Timed out. \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {x}{a}-1}\,\sqrt {\frac {x}{a}+1}+\frac {x}{a}\right ) \,d x \]