Integrand size = 12, antiderivative size = 98 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {x^{1+m-p}}{a (1+m-p)}+\frac {x^{1+m} \sqrt {-1+\frac {x^{-p}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1+m}{2 p},1-\frac {1+m}{2 p},\frac {x^{-2 p}}{a^2}\right )}{(1+m) \sqrt {1-\frac {x^{-p}}{a}}} \] Output:
x^(1+m-p)/a/(1+m-p)+x^(1+m)*(-1+1/a/(x^p))^(1/2)*hypergeom([-1/2, -1/2*(1+ m)/p],[1-1/2*(1+m)/p],1/a^2/(x^(2*p)))/(1+m)/(1-1/a/(x^p))^(1/2)
Time = 4.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.90 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {2^{\frac {1+m}{p}} e^{\text {sech}^{-1}\left (a x^p\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^p\right )}}\right )^{\frac {1+m}{p}} x^{1+m} \left (a x^p\right )^{-\frac {1+m}{p}} \left (-e^{2 \text {sech}^{-1}\left (a x^p\right )} (1+m+p) \operatorname {Hypergeometric2F1}\left (1,-\frac {1+m-3 p}{2 p},\frac {1+m+5 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )+(1+m+3 p) \operatorname {Hypergeometric2F1}\left (1,1-\frac {1+m+p}{2 p},\frac {1+m+3 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )\right )}{(1+m+p) (1+m+3 p)} \] Input:
Integrate[E^ArcSech[a*x^p]*x^m,x]
Output:
(2^((1 + m)/p)*E^ArcSech[a*x^p]*(E^ArcSech[a*x^p]/(1 + E^(2*ArcSech[a*x^p] )))^((1 + m)/p)*x^(1 + m)*(-(E^(2*ArcSech[a*x^p])*(1 + m + p)*Hypergeometr ic2F1[1, -1/2*(1 + m - 3*p)/p, (1 + m + 5*p)/(2*p), -E^(2*ArcSech[a*x^p])] ) + (1 + m + 3*p)*Hypergeometric2F1[1, 1 - (1 + m + p)/(2*p), (1 + m + 3*p )/(2*p), -E^(2*ArcSech[a*x^p])]))/((1 + m + p)*(1 + m + 3*p)*(a*x^p)^((1 + m)/p))
Time = 0.57 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6889, 15, 791, 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.
\(\displaystyle \int x^m e^{\text {sech}^{-1}\left (a x^p\right )} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle \frac {p \int x^{m-p}dx}{a (m+1)}+\frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{m-p}}{\sqrt {1-a x^p} \sqrt {a x^p+1}}dx}{a (m+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{m-p}}{\sqrt {1-a x^p} \sqrt {a x^p+1}}dx}{a (m+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1}\) |
\(\Big \downarrow \) 791 |
\(\displaystyle \frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{m-p}}{\sqrt {1-a^2 x^{2 p}}}dx}{a (m+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} x^{m-p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m-p+1}{2 p},\frac {m+p+1}{2 p},a^2 x^{2 p}\right )}{a (m+1) (m-p+1)}+\frac {p x^{m-p+1}}{a (m+1) (m-p+1)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^p\right )}}{m+1}\) |
Input:
Int[E^ArcSech[a*x^p]*x^m,x]
Output:
(E^ArcSech[a*x^p]*x^(1 + m))/(1 + m) + (p*x^(1 + m - p))/(a*(1 + m)*(1 + m - p)) + (p*x^(1 + m - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Hypergeom etric2F1[1/2, (1 + m - p)/(2*p), (1 + m + p)/(2*p), a^2*x^(2*p)])/(a*(1 + m)*(1 + m - p))
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[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]
\[\int \left (\frac {x^{-p}}{a}+\sqrt {-1+\frac {x^{-p}}{a}}\, \sqrt {1+\frac {x^{-p}}{a}}\right ) x^{m}d x\]
Input:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x)
Output:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x)
Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x, algo rithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {\int x^{m} x^{- p}\, dx + \int a x^{m} \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}\, dx}{a} \] Input:
integrate((1/a/(x**p)+(-1+1/a/(x**p))**(1/2)*(1+1/a/(x**p))**(1/2))*x**m,x )
Output:
(Integral(x**m/x**p, x) + Integral(a*x**m*sqrt(-1 + 1/(a*x**p))*sqrt(1 + 1 /(a*x**p)), x))/a
Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\text {Exception raised: ValueError} \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x, algo rithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(m-p>0)', see `assume?` for more details)Is
\[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}\right )} \,d x } \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x, algo rithm="giac")
Output:
integrate(x^m*(sqrt(1/(a*x^p) + 1)*sqrt(1/(a*x^p) - 1) + 1/(a*x^p)), x)
Timed out. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\int x^m\,\left (\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}\right ) \,d x \] Input:
int(x^m*((1/(a*x^p) - 1)^(1/2)*(1/(a*x^p) + 1)^(1/2) + 1/(a*x^p)),x)
Output:
int(x^m*((1/(a*x^p) - 1)^(1/2)*(1/(a*x^p) + 1)^(1/2) + 1/(a*x^p)), x)
\[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}\, m x -x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}\, p x +x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}\, x +x^{m} m x +x^{m} x -x^{p} \left (\int \frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{3 p} a^{2} m +x^{3 p} a^{2}-x^{p} m -x^{p}}d x \right ) m^{2} p +x^{p} \left (\int \frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{3 p} a^{2} m +x^{3 p} a^{2}-x^{p} m -x^{p}}d x \right ) m \,p^{2}-2 x^{p} \left (\int \frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{3 p} a^{2} m +x^{3 p} a^{2}-x^{p} m -x^{p}}d x \right ) m p +x^{p} \left (\int \frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{3 p} a^{2} m +x^{3 p} a^{2}-x^{p} m -x^{p}}d x \right ) p^{2}-x^{p} \left (\int \frac {x^{m} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{3 p} a^{2} m +x^{3 p} a^{2}-x^{p} m -x^{p}}d x \right ) p}{x^{p} a \left (m^{2}-m p +2 m -p +1\right )} \] Input:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))*x^m,x)
Output:
(x**m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1)*m*x - x**m*sqrt(x**p*a + 1)*sqr t( - x**p*a + 1)*p*x + x**m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1)*x + x**m* m*x + x**m*x - x**p*int((x**m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(3 *p)*a**2*m + x**(3*p)*a**2 - x**p*m - x**p),x)*m**2*p + x**p*int((x**m*sqr t(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(3*p)*a**2*m + x**(3*p)*a**2 - x**p *m - x**p),x)*m*p**2 - 2*x**p*int((x**m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(3*p)*a**2*m + x**(3*p)*a**2 - x**p*m - x**p),x)*m*p + x**p*int((x **m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(3*p)*a**2*m + x**(3*p)*a**2 - x**p*m - x**p),x)*p**2 - x**p*int((x**m*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(3*p)*a**2*m + x**(3*p)*a**2 - x**p*m - x**p),x)*p)/(x**p*a*(m* *2 - m*p + 2*m - p + 1))