Integrand size = 12, antiderivative size = 78 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\frac {x^{3+m}}{a (3+m)}+\frac {x^{1+m} \sqrt {-1+\frac {x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{4},\frac {5+m}{4},\frac {x^4}{a^2}\right )}{(1+m) \sqrt {1-\frac {x^2}{a}}} \] Output:
x^(3+m)/a/(3+m)+x^(1+m)*(-1+x^2/a)^(1/2)*hypergeom([-1/2, 1/4+1/4*m],[5/4+ 1/4*m],x^4/a^2)/(1+m)/(1-x^2/a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(161\) vs. \(2(78)=156\).
Time = 2.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.06 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\frac {2^{\frac {1}{2} (-1-m)} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} \left (\frac {e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{1+e^{2 \text {sech}^{-1}\left (\frac {a}{x^2}\right )}}\right )^{\frac {1}{2} (-1-m)} \left (\frac {a}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \left ((-5+m) \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {5-m}{4},-e^{2 \text {sech}^{-1}\left (\frac {a}{x^2}\right )}\right )-e^{2 \text {sech}^{-1}\left (\frac {a}{x^2}\right )} (-1+m) \operatorname {Hypergeometric2F1}\left (1,\frac {7+m}{4},\frac {9-m}{4},-e^{2 \text {sech}^{-1}\left (\frac {a}{x^2}\right )}\right )\right )}{(-5+m) (-1+m)} \] Input:
Integrate[E^ArcSech[a/x^2]*x^m,x]
Output:
(2^((-1 - m)/2)*E^ArcSech[a/x^2]*(E^ArcSech[a/x^2]/(1 + E^(2*ArcSech[a/x^2 ])))^((-1 - m)/2)*(a/x^2)^((1 + m)/2)*x^(1 + m)*((-5 + m)*Hypergeometric2F 1[1, (3 + m)/4, (5 - m)/4, -E^(2*ArcSech[a/x^2])] - E^(2*ArcSech[a/x^2])*( -1 + m)*Hypergeometric2F1[1, (7 + m)/4, (9 - m)/4, -E^(2*ArcSech[a/x^2])]) )/((-5 + m)*(-1 + m))
Time = 0.59 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45, 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, 888}
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.
| \(\displaystyle \int x^m e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 6889 |
| \(\displaystyle -\frac {2 \int x^{m+2}dx}{a (m+1)}-\frac {2 \sqrt {\frac {1}{\frac {a}{x^2}+1}} \sqrt {\frac {a}{x^2}+1} \int \frac {x^{m+2}}{\sqrt {1-\frac {a}{x^2}} \sqrt {\frac {a}{x^2}+1}}dx}{a (m+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{m+1}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {2 \sqrt {\frac {1}{\frac {a}{x^2}+1}} \sqrt {\frac {a}{x^2}+1} \int \frac {x^{m+2}}{\sqrt {1-\frac {a}{x^2}} \sqrt {\frac {a}{x^2}+1}}dx}{a (m+1)}-\frac {2 x^{m+3}}{a (m+1) (m+3)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{m+1}\) |
| \(\Big \downarrow \) 791 |
| \(\displaystyle -\frac {2 \sqrt {\frac {1}{\frac {a}{x^2}+1}} \sqrt {\frac {a}{x^2}+1} \int \frac {x^{m+2}}{\sqrt {1-\frac {a^2}{x^4}}}dx}{a (m+1)}-\frac {2 x^{m+3}}{a (m+1) (m+3)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{m+1}\) |
| \(\Big \downarrow \) 862 |
| \(\displaystyle \frac {2 \sqrt {\frac {1}{\frac {a}{x^2}+1}} \sqrt {\frac {a}{x^2}+1} \left (\frac {1}{x}\right )^m x^m \int \frac {\left (\frac {1}{x}\right )^{-m-4}}{\sqrt {1-\frac {a^2}{x^4}}}d\frac {1}{x}}{a (m+1)}-\frac {2 x^{m+3}}{a (m+1) (m+3)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{m+1}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle -\frac {2 \sqrt {\frac {1}{\frac {a}{x^2}+1}} \sqrt {\frac {a}{x^2}+1} x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-m-3),\frac {1-m}{4},\frac {a^2}{x^4}\right )}{a (m+1) (m+3)}-\frac {2 x^{m+3}}{a (m+1) (m+3)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )}}{m+1}\) |
Input:
Int[E^ArcSech[a/x^2]*x^m,x]
Output:
(E^ArcSech[a/x^2]*x^(1 + m))/(1 + m) - (2*x^(3 + m))/(a*(1 + m)*(3 + m)) - (2*Sqrt[(1 + a/x^2)^(-1)]*Sqrt[1 + a/x^2]*x^(3 + m)*Hypergeometric2F1[1/2 , (-3 - m)/4, (1 - m)/4, a^2/x^4])/(a*(1 + m)*(3 + m))
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_.)*((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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
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]
\[\int \left (\frac {x^{2}}{a}+\sqrt {-1+\frac {x^{2}}{a}}\, \sqrt {\frac {x^{2}}{a}+1}\right ) x^{m}d x\]
Input:
int((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x)
Output:
int((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x)
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\int { x^{m} {\left (\frac {x^{2}}{a} + \sqrt {\frac {x^{2}}{a} + 1} \sqrt {\frac {x^{2}}{a} - 1}\right )} \,d x } \] Input:
integrate((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x, algorithm="frica s")
Output:
integral((x^2*x^m + a*x^m*sqrt((x^2 + a)/a)*sqrt((x^2 - a)/a))/a, x)
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\frac {\int x^{2} x^{m}\, dx + \int a x^{m} \sqrt {-1 + \frac {x^{2}}{a}} \sqrt {1 + \frac {x^{2}}{a}}\, dx}{a} \] Input:
integrate((x**2/a+(-1+x**2/a)**(1/2)*(x**2/a+1)**(1/2))*x**m,x)
Output:
(Integral(x**2*x**m, x) + Integral(a*x**m*sqrt(-1 + x**2/a)*sqrt(1 + x**2/ a), x))/a
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\int { x^{m} {\left (\frac {x^{2}}{a} + \sqrt {\frac {x^{2}}{a} + 1} \sqrt {\frac {x^{2}}{a} - 1}\right )} \,d x } \] Input:
integrate((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x, algorithm="maxim a")
Output:
x^3*x^m/(a*(m + 3)) + integrate(sqrt(x^2 + a)*sqrt(x^2 - a)*x^m, x)/a
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\int { x^{m} {\left (\frac {x^{2}}{a} + \sqrt {\frac {x^{2}}{a} + 1} \sqrt {\frac {x^{2}}{a} - 1}\right )} \,d x } \] Input:
integrate((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x, algorithm="giac" )
Output:
integrate(x^m*(x^2/a + sqrt(x^2/a + 1)*sqrt(x^2/a - 1)), x)
Timed out. \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {x^2}{a}-1}\,\sqrt {\frac {x^2}{a}+1}+\frac {x^2}{a}\right ) \,d x \] Input:
int(x^m*((x^2/a - 1)^(1/2)*(x^2/a + 1)^(1/2) + x^2/a),x)
Output:
int(x^m*((x^2/a - 1)^(1/2)*(x^2/a + 1)^(1/2) + x^2/a), x)
\[ \int e^{\text {sech}^{-1}\left (\frac {a}{x^2}\right )} x^m \, dx=\frac {x^{m} \sqrt {x^{2}+a}\, \sqrt {x^{2}-a}\, x +x^{m} x^{3}+2 \left (\int \frac {x^{m} \sqrt {x^{2}+a}\, \sqrt {x^{2}-a}}{-m \,x^{4}-3 x^{4}+a^{2} m +3 a^{2}}d x \right ) a^{2} m +6 \left (\int \frac {x^{m} \sqrt {x^{2}+a}\, \sqrt {x^{2}-a}}{-m \,x^{4}-3 x^{4}+a^{2} m +3 a^{2}}d x \right ) a^{2}}{a \left (m +3\right )} \] Input:
int((x^2/a+(-1+x^2/a)^(1/2)*(x^2/a+1)^(1/2))*x^m,x)
Output:
(x**m*sqrt(a + x**2)*sqrt( - a + x**2)*x + x**m*x**3 + 2*int((x**m*sqrt(a + x**2)*sqrt( - a + x**2))/(a**2*m + 3*a**2 - m*x**4 - 3*x**4),x)*a**2*m + 6*int((x**m*sqrt(a + x**2)*sqrt( - a + x**2))/(a**2*m + 3*a**2 - m*x**4 - 3*x**4),x)*a**2)/(a*(m + 3))