Integrand size = 10, antiderivative size = 163 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=-\frac {1}{8 a x^8}-\frac {\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 x^7}+\frac {a^2 \sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{48 x^5}+\frac {5 a^4 \sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{192 x^3}+\frac {5 a^6 \sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{128 x}+\frac {5}{128} a^7 \text {sech}^{-1}(a x) \] Output:
-1/8/a/x^8-1/8*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)/x^7+1/48*a^2*(-1+1/a/x)^(1 /2)*(1+1/a/x)^(1/2)/x^5+5/192*a^4*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)/x^3+5/1 28*a^6*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)/x+5/128*a^7*arcsech(a*x)
Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {-48+\sqrt {\frac {1-a x}{1+a x}} \left (-48-48 a x+8 a^2 x^2+8 a^3 x^3+10 a^4 x^4+10 a^5 x^5+15 a^6 x^6+15 a^7 x^7\right )-15 a^8 x^8 \log (x)+15 a^8 x^8 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{384 a x^8} \] Input:
Integrate[E^ArcSech[a*x]/x^8,x]
Output:
(-48 + Sqrt[(1 - a*x)/(1 + a*x)]*(-48 - 48*a*x + 8*a^2*x^2 + 8*a^3*x^3 + 1 0*a^4*x^4 + 10*a^5*x^5 + 15*a^6*x^6 + 15*a^7*x^7) - 15*a^8*x^8*Log[x] + 15 *a^8*x^8*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)] ])/(384*a*x^8)
Time = 0.58 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6889, 15, 114, 27, 114, 27, 114, 27, 114, 25, 27, 103, 221}
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.
| \(\displaystyle \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx\) |
| \(\Big \downarrow \) 6889 |
| \(\displaystyle -\frac {\int \frac {1}{x^9}dx}{7 a}-\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^9 \sqrt {1-a x} \sqrt {a x+1}}dx}{7 a}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^9 \sqrt {1-a x} \sqrt {a x+1}}dx}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{8} \int -\frac {7 a^2}{x^7 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \int \frac {1}{x^7 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (-\frac {1}{6} \int -\frac {5 a^2}{x^5 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \int \frac {1}{x^5 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (-\frac {1}{4} \int -\frac {3 a^2}{x^3 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (-\frac {1}{2} \int -\frac {a^2}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} \int \frac {a^2}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {1}{x \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (-\frac {1}{2} a^3 \int \frac {1}{a-a (1-a x) (a x+1)}d\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {7}{8} a^2 \left (\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (-\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{4 x^4}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{6 x^6}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{8 x^8}\right )}{7 a}+\frac {1}{56 a x^8}-\frac {e^{\text {sech}^{-1}(a x)}}{7 x^7}\) |
Input:
Int[E^ArcSech[a*x]/x^8,x]
Output:
1/(56*a*x^8) - E^ArcSech[a*x]/(7*x^7) - (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x ]*(-1/8*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^8 + (7*a^2*(-1/6*(Sqrt[1 - a*x]*Sq rt[1 + a*x])/x^6 + (5*a^2*(-1/4*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^4 + (3*a^2 *(-1/2*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^2 - (a^2*ArcTanh[Sqrt[1 - a*x]*Sqrt [1 + a*x]])/2))/4))/6))/8))/(7*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[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{8} x^{8}+15 a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}+10 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}+8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-48 \sqrt {-a^{2} x^{2}+1}\right )}{384 x^{7} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{8 a \,x^{8}}\) | \(152\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^8,x,method=_RETURNVERBOSE)
Output:
1/384*(-(a*x-1)/a/x)^(1/2)/x^7*((a*x+1)/a/x)^(1/2)*(15*arctanh(1/(-a^2*x^2 +1)^(1/2))*a^8*x^8+15*a^6*x^6*(-a^2*x^2+1)^(1/2)+10*a^4*x^4*(-a^2*x^2+1)^( 1/2)+8*(-a^2*x^2+1)^(1/2)*a^2*x^2-48*(-a^2*x^2+1)^(1/2))/(-a^2*x^2+1)^(1/2 )-1/8/a/x^8
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - 15 \, a^{8} x^{8} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, {\left (15 \, a^{7} x^{7} + 10 \, a^{5} x^{5} + 8 \, a^{3} x^{3} - 48 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 96}{768 \, a x^{8}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^8,x, algorithm="frica s")
Output:
1/768*(15*a^8*x^8*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) + 1 ) - 15*a^8*x^8*log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*(15*a^7*x^7 + 10*a^5*x^5 + 8*a^3*x^3 - 48*a*x)*sqrt((a*x + 1)/(a*x))*sq rt(-(a*x - 1)/(a*x)) - 96)/(a*x^8)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {\int \frac {1}{x^{9}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{8}}\, dx}{a} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))/x**8,x)
Output:
(Integral(x**(-9), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x* *8, x))/a
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{8}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^8,x, algorithm="maxim a")
Output:
integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^9, x)/a - 1/8/(a*x^8)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{8}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^8,x, algorithm="giac" )
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^8, x)
Time = 93.72 (sec) , antiderivative size = 1155, normalized size of antiderivative = 7.09 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\text {Too large to display} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^8,x)
Output:
(5*a^7*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/32 - ( (1723*a^7*((1/(a*x) - 1)^(1/2) - 1i)^5)/(96*((1/(a*x) + 1)^(1/2) - 1)^5) - (235*a^7*((1/(a*x) - 1)^(1/2) - 1i)^3)/(96*((1/(a*x) + 1)^(1/2) - 1)^3) + (72283*a^7*((1/(a*x) - 1)^(1/2) - 1i)^7)/(32*((1/(a*x) + 1)^(1/2) - 1)^7) + (848801*a^7*((1/(a*x) - 1)^(1/2) - 1i)^9)/(32*((1/(a*x) + 1)^(1/2) - 1) ^9) + (4181067*a^7*((1/(a*x) - 1)^(1/2) - 1i)^11)/(32*((1/(a*x) + 1)^(1/2) - 1)^11) + (10994181*a^7*((1/(a*x) - 1)^(1/2) - 1i)^13)/(32*((1/(a*x) + 1 )^(1/2) - 1)^13) + (17457599*a^7*((1/(a*x) - 1)^(1/2) - 1i)^15)/(32*((1/(a *x) + 1)^(1/2) - 1)^15) + (17457599*a^7*((1/(a*x) - 1)^(1/2) - 1i)^17)/(32 *((1/(a*x) + 1)^(1/2) - 1)^17) + (10994181*a^7*((1/(a*x) - 1)^(1/2) - 1i)^ 19)/(32*((1/(a*x) + 1)^(1/2) - 1)^19) + (4181067*a^7*((1/(a*x) - 1)^(1/2) - 1i)^21)/(32*((1/(a*x) + 1)^(1/2) - 1)^21) + (848801*a^7*((1/(a*x) - 1)^( 1/2) - 1i)^23)/(32*((1/(a*x) + 1)^(1/2) - 1)^23) + (72283*a^7*((1/(a*x) - 1)^(1/2) - 1i)^25)/(32*((1/(a*x) + 1)^(1/2) - 1)^25) + (1723*a^7*((1/(a*x) - 1)^(1/2) - 1i)^27)/(96*((1/(a*x) + 1)^(1/2) - 1)^27) - (235*a^7*((1/(a* x) - 1)^(1/2) - 1i)^29)/(96*((1/(a*x) + 1)^(1/2) - 1)^29) + (5*a^7*((1/(a* x) - 1)^(1/2) - 1i)^31)/(32*((1/(a*x) + 1)^(1/2) - 1)^31) + (5*a^7*((1/(a* x) - 1)^(1/2) - 1i))/(32*((1/(a*x) + 1)^(1/2) - 1)))/((120*((1/(a*x) - 1)^ (1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 - (16*((1/(a*x) - 1)^(1/2) - 1i )^2)/((1/(a*x) + 1)^(1/2) - 1)^2 - (560*((1/(a*x) - 1)^(1/2) - 1i)^6)/(...
Time = 0.18 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^8} \, dx=\frac {15 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{6} x^{6}+10 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{4} x^{4}+8 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{2} x^{2}-48 \sqrt {a x +1}\, \sqrt {-a x +1}+15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{8} x^{8}-15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{8} x^{8}+15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{8} x^{8}-15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{8} x^{8}-48}{384 a \,x^{8}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^8,x)
Output:
(15*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**6*x**6 + 10*sqrt(a*x + 1)*sqrt( - a* x + 1)*a**4*x**4 + 8*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**2*x**2 - 48*sqrt(a* x + 1)*sqrt( - a*x + 1) + 15*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sq rt(2))/2) - 1)*a**8*x**8 - 15*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/s qrt(2))/2) + 1)*a**8*x**8 + 15*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqr t(2))/2) - 1)*a**8*x**8 - 15*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt( 2))/2) + 1)*a**8*x**8 - 48)/(384*a*x**8)