Integrand size = 12, antiderivative size = 77 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=-\frac {x^{-2-p}}{a (2+p)}-\frac {\sqrt {-1+\frac {x^{-p}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{p},1+\frac {1}{p},\frac {x^{-2 p}}{a^2}\right )}{2 x^2 \sqrt {1-\frac {x^{-p}}{a}}} \] Output:
-x^(-2-p)/a/(2+p)-1/2*(-1+1/a/(x^p))^(1/2)*hypergeom([-1/2, 1/p],[1+1/p],1 /a^2/(x^(2*p)))/x^2/(1-1/a/(x^p))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(77)=154\).
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\frac {x^{-2-p} \left (-1-\sqrt {\frac {1-a x^p}{1+a x^p}}-a x^p \sqrt {\frac {1-a x^p}{1+a x^p}}+\frac {a^2 p x^{2 p} \sqrt {\frac {1-a x^p}{1+a x^p}} \sqrt {1-a^2 x^{2 p}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {1}{p},\frac {3}{2}-\frac {1}{p},a^2 x^{2 p}\right )}{(-2+p) \left (-1+a x^p\right )}\right )}{a (2+p)} \] Input:
Integrate[E^ArcSech[a*x^p]/x^3,x]
Output:
(x^(-2 - p)*(-1 - Sqrt[(1 - a*x^p)/(1 + a*x^p)] - a*x^p*Sqrt[(1 - a*x^p)/( 1 + a*x^p)] + (a^2*p*x^(2*p)*Sqrt[(1 - a*x^p)/(1 + a*x^p)]*Sqrt[1 - a^2*x^ (2*p)]*Hypergeometric2F1[1/2, 1/2 - p^(-1), 3/2 - p^(-1), a^2*x^(2*p)])/(( -2 + p)*(-1 + a*x^p))))/(a*(2 + p))
Time = 0.51 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.48, 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.
\(\displaystyle \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle -\frac {p \int x^{-p-3}dx}{2 a}-\frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{-p-3}}{\sqrt {1-a x^p} \sqrt {a x^p+1}}dx}{2 a}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{-p-3}}{\sqrt {1-a x^p} \sqrt {a x^p+1}}dx}{2 a}+\frac {p x^{-p-2}}{2 a (p+2)}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{2 x^2}\) |
\(\Big \downarrow \) 791 |
\(\displaystyle -\frac {p \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \int \frac {x^{-p-3}}{\sqrt {1-a^2 x^{2 p}}}dx}{2 a}+\frac {p x^{-p-2}}{2 a (p+2)}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{2 x^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {p x^{-p-2} \sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {p+2}{2 p},\frac {1}{2} \left (1-\frac {2}{p}\right ),a^2 x^{2 p}\right )}{2 a (p+2)}+\frac {p x^{-p-2}}{2 a (p+2)}-\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{2 x^2}\) |
Input:
Int[E^ArcSech[a*x^p]/x^3,x]
Output:
-1/2*E^ArcSech[a*x^p]/x^2 + (p*x^(-2 - p))/(2*a*(2 + p)) + (p*x^(-2 - p)*S qrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Hypergeometric2F1[1/2, -1/2*(2 + p)/ p, (1 - 2/p)/2, a^2*x^(2*p)])/(2*a*(2 + 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]
\[\int \frac {\frac {x^{-p}}{a}+\sqrt {-1+\frac {x^{-p}}{a}}\, \sqrt {1+\frac {x^{-p}}{a}}}{x^{3}}d x\]
Input:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x)
Output:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x)
Exception generated. \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x, algo rithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\frac {\int \frac {x^{- p}}{x^{3}}\, dx + \int \frac {a \sqrt {-1 + \frac {x^{- p}}{a}} \sqrt {1 + \frac {x^{- p}}{a}}}{x^{3}}\, dx}{a} \] Input:
integrate((1/a/(x**p)+(-1+1/a/(x**p))**(1/2)*(1+1/a/(x**p))**(1/2))/x**3,x )
Output:
(Integral(1/(x**3*x**p), x) + Integral(a*sqrt(-1 + 1/(a*x**p))*sqrt(1 + 1/ (a*x**p))/x**3, x))/a
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x^{3}} \,d x } \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x, algo rithm="maxima")
Output:
integrate(sqrt(a*x^p + 1)*sqrt(-a*x^p + 1)/(x^3*x^p), x)/a - x^(-p - 2)/(a *(p + 2))
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{p}} + 1} \sqrt {\frac {1}{a x^{p}} - 1} + \frac {1}{a x^{p}}}{x^{3}} \,d x } \] Input:
integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x, algo rithm="giac")
Output:
integrate((sqrt(1/(a*x^p) + 1)*sqrt(1/(a*x^p) - 1) + 1/(a*x^p))/x^3, x)
Timed out. \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\int \frac {\sqrt {\frac {1}{a\,x^p}-1}\,\sqrt {\frac {1}{a\,x^p}+1}+\frac {1}{a\,x^p}}{x^3} \,d x \] Input:
int(((1/(a*x^p) - 1)^(1/2)*(1/(a*x^p) + 1)^(1/2) + 1/(a*x^p))/x^3,x)
Output:
int(((1/(a*x^p) - 1)^(1/2)*(1/(a*x^p) + 1)^(1/2) + 1/(a*x^p))/x^3, x)
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x^3} \, dx=\frac {-\sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}+x^{p} \left (\int \frac {x^{p} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{2 p} a^{2} p \,x^{3}+2 x^{2 p} a^{2} x^{3}-p \,x^{3}-2 x^{3}}d x \right ) a^{2} p^{2} x^{2}+2 x^{p} \left (\int \frac {x^{p} \sqrt {x^{p} a +1}\, \sqrt {-x^{p} a +1}}{x^{2 p} a^{2} p \,x^{3}+2 x^{2 p} a^{2} x^{3}-p \,x^{3}-2 x^{3}}d x \right ) a^{2} p \,x^{2}-1}{x^{p} a \,x^{2} \left (p +2\right )} \] Input:
int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x^3,x)
Output:
( - sqrt(x**p*a + 1)*sqrt( - x**p*a + 1) + x**p*int((x**p*sqrt(x**p*a + 1) *sqrt( - x**p*a + 1))/(x**(2*p)*a**2*p*x**3 + 2*x**(2*p)*a**2*x**3 - p*x** 3 - 2*x**3),x)*a**2*p**2*x**2 + 2*x**p*int((x**p*sqrt(x**p*a + 1)*sqrt( - x**p*a + 1))/(x**(2*p)*a**2*p*x**3 + 2*x**(2*p)*a**2*x**3 - p*x**3 - 2*x** 3),x)*a**2*p*x**2 - 1)/(x**p*a*x**2*(p + 2))