Integrand size = 12, antiderivative size = 175 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{36 x^6}+\frac {5 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{144 x^4}+\frac {5 b c^4 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{96 x^2}-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}+\frac {5 b c^6 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{96 \sqrt {1-c^2 x^2}} \] Output:
1/36*b*((-c*x+1)/(c*x+1))^(1/2)*(c*x+1)/x^6+5/144*b*c^2*((-c*x+1)/(c*x+1)) ^(1/2)*(c*x+1)/x^4+5/96*b*c^4*((-c*x+1)/(c*x+1))^(1/2)*(c*x+1)/x^2-1/6*(a+ b*arcsech(c*x))/x^6+5/96*b*c^6*((-c*x+1)/(c*x+1))^(1/2)*(c*x+1)*arctanh((- c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)
Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=-\frac {a}{6 x^6}+b \left (\frac {1}{36 x^6}+\frac {c}{36 x^5}+\frac {5 c^2}{144 x^4}+\frac {5 c^3}{144 x^3}+\frac {5 c^4}{96 x^2}+\frac {5 c^5}{96 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{6 x^6}-\frac {5}{96} b c^6 \log (x)+\frac {5}{96} b c^6 \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right ) \] Input:
Integrate[(a + b*ArcSech[c*x])/x^7,x]
Output:
-1/6*a/x^6 + b*(1/(36*x^6) + c/(36*x^5) + (5*c^2)/(144*x^4) + (5*c^3)/(144 *x^3) + (5*c^4)/(96*x^2) + (5*c^5)/(96*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b* ArcSech[c*x])/(6*x^6) - (5*b*c^6*Log[x])/96 + (5*b*c^6*Log[1 + Sqrt[(1 - c *x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/96
Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6837, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx\) |
| \(\Big \downarrow \) 6837 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x^7 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {1}{6} \int -\frac {5 c^2}{x^5 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \int \frac {1}{x^5 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (-\frac {1}{4} \int -\frac {3 c^2}{x^3 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \int \frac {1}{x^3 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (-\frac {1}{2} \int -\frac {c^2}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} \int \frac {c^2}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (-\frac {1}{2} c^3 \int \frac {1}{c-c (1-c x) (c x+1)}d\left (\sqrt {1-c x} \sqrt {c x+1}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )-\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{6 x^6}-\frac {1}{6} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (-\frac {1}{2} c^2 \text {arctanh}\left (\sqrt {1-c x} \sqrt {c x+1}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{4 x^4}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{6 x^6}\right )\) |
Input:
Int[(a + b*ArcSech[c*x])/x^7,x]
Output:
-1/6*(a + b*ArcSech[c*x])/x^6 - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-1/ 6*(Sqrt[1 - c*x]*Sqrt[1 + c*x])/x^6 + (5*c^2*(-1/4*(Sqrt[1 - c*x]*Sqrt[1 + c*x])/x^4 + (3*c^2*(-1/2*(Sqrt[1 - c*x]*Sqrt[1 + c*x])/x^2 - (c^2*ArcTanh [Sqrt[1 - c*x]*Sqrt[1 + c*x]])/2))/4))/6))/6
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[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(d*x)^(m + 1)*((a + b*ArcSech[c*x])/(d*(m + 1))), x] + Simp[b*(Sqrt[1 + c*x]/(m + 1))*Sqrt[1/(1 + c*x)] Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]) , x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86
| method | result | size |
| parts | \(-\frac {a}{6 x^{6}}+b \,c^{6} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\) | \(151\) |
| derivativedivides | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(155\) |
| default | \(c^{6} \left (-\frac {a}{6 c^{6} x^{6}}+b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{6 c^{6} x^{6}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (15 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+10 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\right )}{288 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(155\) |
Input:
int((a+b*arcsech(c*x))/x^7,x,method=_RETURNVERBOSE)
Output:
-1/6*a/x^6+b*c^6*(-1/6/c^6/x^6*arcsech(c*x)+1/288*(-(c*x-1)/c/x)^(1/2)/c^5 /x^5*((c*x+1)/c/x)^(1/2)*(15*arctanh(1/(-c^2*x^2+1)^(1/2))*c^6*x^6+15*(-c^ 2*x^2+1)^(1/2)*c^4*x^4+10*(-c^2*x^2+1)^(1/2)*c^2*x^2+8*(-c^2*x^2+1)^(1/2)) /(-c^2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\frac {3 \, {\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (15 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 48 \, a}{288 \, x^{6}} \] Input:
integrate((a+b*arcsech(c*x))/x^7,x, algorithm="fricas")
Output:
1/288*(3*(5*b*c^6*x^6 - 16*b)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1) /(c*x)) + (15*b*c^5*x^5 + 10*b*c^3*x^3 + 8*b*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2 *x^2)) - 48*a)/x^6
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{7}}\, dx \] Input:
integrate((a+b*asech(c*x))/x**7,x)
Output:
Integral((a + b*asech(c*x))/x**7, x)
Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.06 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\frac {1}{576} \, b {\left (\frac {15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right ) - \frac {2 \, {\left (15 \, c^{12} x^{5} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} - 40 \, c^{10} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, c^{8} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}}{c^{6} x^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac {96 \, \operatorname {arsech}\left (c x\right )}{x^{6}}\right )} - \frac {a}{6 \, x^{6}} \] Input:
integrate((a+b*arcsech(c*x))/x^7,x, algorithm="maxima")
Output:
1/576*b*((15*c^7*log(c*x*sqrt(1/(c^2*x^2) - 1) + 1) - 15*c^7*log(c*x*sqrt( 1/(c^2*x^2) - 1) - 1) - 2*(15*c^12*x^5*(1/(c^2*x^2) - 1)^(5/2) - 40*c^10*x ^3*(1/(c^2*x^2) - 1)^(3/2) + 33*c^8*x*sqrt(1/(c^2*x^2) - 1))/(c^6*x^6*(1/( c^2*x^2) - 1)^3 - 3*c^4*x^4*(1/(c^2*x^2) - 1)^2 + 3*c^2*x^2*(1/(c^2*x^2) - 1) - 1))/c - 96*arcsech(c*x)/x^6) - 1/6*a/x^6
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{x^{7}} \,d x } \] Input:
integrate((a+b*arcsech(c*x))/x^7,x, algorithm="giac")
Output:
integrate((b*arcsech(c*x) + a)/x^7, x)
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^7} \,d x \] Input:
int((a + b*acosh(1/(c*x)))/x^7,x)
Output:
int((a + b*acosh(1/(c*x)))/x^7, x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^7} \, dx=\frac {6 \left (\int \frac {\mathit {asech} \left (c x \right )}{x^{7}}d x \right ) b \,x^{6}-a}{6 x^{6}} \] Input:
int((a+b*asech(c*x))/x^7,x)
Output:
(6*int(asech(c*x)/x**7,x)*b*x**6 - a)/(6*x**6)