Integrand size = 10, antiderivative size = 116 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=-\frac {8}{105} a^6 \left (-1+\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}-\frac {1}{7 a x^7}-\frac {a^2 \left (-1+\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{7 x^4}-\frac {4 a^4 \left (-1+\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{35 x^2} \] Output:
-8/105*a^6*(-1+1/a/x)^(3/2)*(1+1/a/x)^(3/2)-1/7/a/x^7-1/7*a^2*(-1+1/a/x)^( 3/2)*(1+1/a/x)^(3/2)/x^4-4/35*a^4*(-1+1/a/x)^(3/2)*(1+1/a/x)^(3/2)/x^2
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {-15+\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-15+15 a x-12 a^2 x^2+12 a^3 x^3-8 a^4 x^4+8 a^5 x^5\right )}{105 a x^7} \] Input:
Integrate[E^ArcSech[a*x]/x^7,x]
Output:
(-15 + Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-15 + 15*a*x - 12*a^2*x^2 + 12*a^3*x^3 - 8*a^4*x^4 + 8*a^5*x^5))/(105*a*x^7)
Result contains higher order function than in optimal. Order 3 vs. order 2 in optimal.
Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.51, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6889, 15, 114, 27, 114, 27, 114, 27, 106}
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.
\(\displaystyle \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle -\frac {\int \frac {1}{x^8}dx}{6 a}-\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^8 \sqrt {1-a x} \sqrt {a x+1}}dx}{6 a}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^8 \sqrt {1-a x} \sqrt {a x+1}}dx}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{7} \int -\frac {6 a^2}{x^6 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \left (-\frac {1}{5} \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \left (\frac {4}{5} a^2 \left (-\frac {1}{3} \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {6}{7} a^2 \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a x} \sqrt {a x+1}}{3 x}-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{7 x^7}\right )}{6 a}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}\) |
Input:
Int[E^ArcSech[a*x]/x^7,x]
Output:
1/(42*a*x^7) - E^ArcSech[a*x]/(6*x^6) - (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x ]*(-1/7*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^7 + (6*a^2*(-1/5*(Sqrt[1 - a*x]*Sq rt[1 + a*x])/x^5 + (4*a^2*(-1/3*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^3 - (2*a^2 *Sqrt[1 - a*x]*Sqrt[1 + a*x])/(3*x)))/5))/7))/(6*a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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]
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (8 a^{4} x^{4}+12 a^{2} x^{2}+15\right )}{105 x^{6}}-\frac {1}{7 a \,x^{7}}\) | \(71\) |
orering | \(\frac {\left (\frac {64}{105} a^{6} x^{7}-\frac {32}{105} x^{5} a^{4}-\frac {8}{105} a^{2} x^{3}-\frac {13}{35} x \right ) \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{x^{7}}+\frac {\left (8 a^{4} x^{4}+4 a^{2} x^{2}+3\right ) x^{2} \left (a x -1\right ) \left (a x +1\right ) \left (\frac {-\frac {1}{a \,x^{2}}-\frac {\sqrt {1+\frac {1}{a x}}}{2 \sqrt {-1+\frac {1}{a x}}\, a \,x^{2}}-\frac {\sqrt {-1+\frac {1}{a x}}}{2 \sqrt {1+\frac {1}{a x}}\, a \,x^{2}}}{x^{7}}-\frac {7 \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{x^{8}}\right )}{105}\) | \(208\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^7,x,method=_RETURNVERBOSE)
Output:
1/105*(-(a*x-1)/a/x)^(1/2)/x^6*((a*x+1)/a/x)^(1/2)*(a^2*x^2-1)*(8*a^4*x^4+ 12*a^2*x^2+15)-1/7/a/x^7
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 15}{105 \, a x^{7}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^7,x, algorithm="frica s")
Output:
1/105*((8*a^7*x^7 + 4*a^5*x^5 + 3*a^3*x^3 - 15*a*x)*sqrt((a*x + 1)/(a*x))* sqrt(-(a*x - 1)/(a*x)) - 15)/(a*x^7)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {\int \frac {1}{x^{8}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{7}}\, dx}{a} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))/x**7,x)
Output:
(Integral(x**(-8), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x* *7, x))/a
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {{\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, a x^{8}} - \frac {1}{7 \, a x^{7}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^7,x, algorithm="maxim a")
Output:
1/105*(8*a^6*x^7 + 4*a^4*x^5 + 3*a^2*x^3 - 15*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a*x^8) - 1/7/(a*x^7)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{7}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^7,x, algorithm="giac" )
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^7, x)
Time = 23.96 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{35}-\frac {\sqrt {\frac {1}{a\,x}+1}}{7}+\frac {4\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{105}+\frac {8\,a^6\,x^6\,\sqrt {\frac {1}{a\,x}+1}}{105}\right )}{x^6}-\frac {1}{7\,a\,x^7} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^7,x)
Output:
((1/(a*x) - 1)^(1/2)*((a^2*x^2*(1/(a*x) + 1)^(1/2))/35 - (1/(a*x) + 1)^(1/ 2)/7 + (4*a^4*x^4*(1/(a*x) + 1)^(1/2))/105 + (8*a^6*x^6*(1/(a*x) + 1)^(1/2 ))/105))/x^6 - 1/(7*a*x^7)
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx=\frac {8 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{6} x^{6}+4 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{4} x^{4}+3 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{2} x^{2}-15 \sqrt {a x +1}\, \sqrt {-a x +1}-15}{105 a \,x^{7}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^7,x)
Output:
(8*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**6*x**6 + 4*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**4*x**4 + 3*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**2*x**2 - 15*sqrt(a*x + 1)*sqrt( - a*x + 1) - 15)/(105*a*x**7)