Integrand size = 12, antiderivative size = 109 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=\frac {b \sqrt {1-c x}}{25 x^5 \sqrt {\frac {1}{1+c x}}}+\frac {4 b c^2 \sqrt {1-c x}}{75 x^3 \sqrt {\frac {1}{1+c x}}}+\frac {8 b c^4 \sqrt {1-c x}}{75 x \sqrt {\frac {1}{1+c x}}}-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5} \]
1/5*(-a-b*arcsech(c*x))/x^5+1/25*b*(-c*x+1)^(1/2)/x^5/(1/(c*x+1))^(1/2)+4/ 75*b*c^2*(-c*x+1)^(1/2)/x^3/(1/(c*x+1))^(1/2)+8/75*b*c^4*(-c*x+1)^(1/2)/x/ (1/(c*x+1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=-\frac {a}{5 x^5}+b \left (\frac {8 c^5}{75}+\frac {1}{25 x^5}+\frac {c}{25 x^4}+\frac {4 c^2}{75 x^3}+\frac {4 c^3}{75 x^2}+\frac {8 c^4}{75 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b \text {sech}^{-1}(c x)}{5 x^5} \]
-1/5*a/x^5 + b*((8*c^5)/75 + 1/(25*x^5) + c/(25*x^4) + (4*c^2)/(75*x^3) + (4*c^3)/(75*x^2) + (8*c^4)/(75*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[ c*x])/(5*x^5)
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6837, 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.
\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx\) |
\(\Big \downarrow \) 6837 |
\(\displaystyle -\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x^6 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {1}{5} \int -\frac {4 c^2}{x^4 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{5 x^5}\right )-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4}{5} c^2 \int \frac {1}{x^4 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{5 x^5}\right )-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4}{5} c^2 \left (-\frac {1}{3} \int -\frac {2 c^2}{x^2 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{3 x^3}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{5 x^5}\right )-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4}{5} c^2 \left (\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{3 x^3}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{5 x^5}\right )-\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{5 x^5}-\frac {1}{5} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {4}{5} c^2 \left (-\frac {2 c^2 \sqrt {1-c x} \sqrt {c x+1}}{3 x}-\frac {\sqrt {1-c x} \sqrt {c x+1}}{3 x^3}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{5 x^5}\right )\) |
-1/5*(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-1/5*(Sqrt[1 - c*x]*Sqrt[1 + c *x])/x^5 + (4*c^2*(-1/3*(Sqrt[1 - c*x]*Sqrt[1 + c*x])/x^3 - (2*c^2*Sqrt[1 - c*x]*Sqrt[1 + c*x])/(3*x)))/5)) - (a + b*ArcSech[c*x])/(5*x^5)
3.1.31.3.1 Defintions of rubi rules used
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[((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 = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74
method | result | size |
parts | \(-\frac {a}{5 x^{5}}+b \,c^{5} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\) | \(81\) |
derivativedivides | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\right )\) | \(85\) |
default | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (8 c^{4} x^{4}+4 c^{2} x^{2}+3\right )}{75 c^{4} x^{4}}\right )\right )\) | \(85\) |
-1/5*a/x^5+b*c^5*(-1/5/c^5/x^5*arcsech(c*x)+1/75*(-(c*x-1)/c/x)^(1/2)/c^4/ x^4*((c*x+1)/c/x)^(1/2)*(8*c^4*x^4+4*c^2*x^2+3))
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=-\frac {15 \, b \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{5} x^{5} + 4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 15 \, a}{75 \, x^{5}} \]
-1/75*(15*b*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (8*b*c^5 *x^5 + 4*b*c^3*x^3 + 3*b*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 15*a)/x^5
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{6}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.67 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=\frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \]
1/75*b*((3*c^6*(1/(c^2*x^2) - 1)^(5/2) + 10*c^6*(1/(c^2*x^2) - 1)^(3/2) + 15*c^6*sqrt(1/(c^2*x^2) - 1))/c - 15*arcsech(c*x)/x^5) - 1/5*a/x^5
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^6} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \]