Integrand size = 14, antiderivative size = 122 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=-\frac {2 b^2}{27 x^3}-\frac {4 b^2 c^2}{9 x}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {4 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3} \]
-2/27*b^2/x^3-4/9*b^2*c^2/x-1/3*(a+b*arcsech(c*x))^2/x^3+2/9*b*(c*x+1)*(a+ b*arcsech(c*x))*((-c*x+1)/(c*x+1))^(1/2)/x^3+4/9*b*c^2*(c*x+1)*(a+b*arcsec h(c*x))*((-c*x+1)/(c*x+1))^(1/2)/x
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=\frac {-9 a^2-2 b^2 \left (1+6 c^2 x^2\right )+6 a b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )+6 b \left (-3 a+b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )\right ) \text {sech}^{-1}(c x)-9 b^2 \text {sech}^{-1}(c x)^2}{27 x^3} \]
(-9*a^2 - 2*b^2*(1 + 6*c^2*x^2) + 6*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 6*b*(-3*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c *x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x] - 9*b^2*ArcSech[c*x]^2)/(27*x^3)
Time = 0.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6839, 5970, 3042, 3791, 3042, 3777, 26, 3042, 26, 3118}
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.
| \(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -c^3 \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^3 x^3}d\text {sech}^{-1}(c x)\) |
| \(\Big \downarrow \) 5970 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int \frac {a+b \text {sech}^{-1}(c x)}{c^3 x^3}d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int \left (a+b \text {sech}^{-1}(c x)\right ) \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \int \frac {a+b \text {sech}^{-1}(c x)}{c x}d\text {sech}^{-1}(c x)+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \int \left (a+b \text {sech}^{-1}(c x)\right ) \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-i b \int -\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}d\text {sech}^{-1}(c x)\right )+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-b \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}d\text {sech}^{-1}(c x)\right )+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-b \int -i \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}+i b \int \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}-\frac {b}{9 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3 x^3}+\frac {2}{3} \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-\frac {b}{c x}\right )-\frac {b}{9 c^3 x^3}\right )\right )\) |
-(c^3*((a + b*ArcSech[c*x])^2/(3*c^3*x^3) - (2*b*(-1/9*b/(c^3*x^3) + (Sqrt [(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(3*c^3*x^3) + (2*(-( b/(c*x)) + (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(c*x )))/3))/3))
3.1.40.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
Time = 0.81 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.57
| method | result | size |
| parts | \(-\frac {a^{2}}{3 x^{3}}+b^{2} c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+2 a b \,c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\) | \(191\) |
| derivativedivides | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(192\) |
| default | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\) | \(192\) |
-1/3*a^2/x^3+b^2*c^3*(-1/3/c^3/x^3*arcsech(c*x)^2+4/9*arcsech(c*x)*(-(c*x- 1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+2/9/c^2/x^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1 )/c/x)^(1/2)*arcsech(c*x)-4/9/c/x-2/27/c^3/x^3)+2*a*b*c^3*(-1/3/c^3/x^3*ar csech(c*x)+1/9*(-(c*x-1)/c/x)^(1/2)/c^2/x^2*((c*x+1)/c/x)^(1/2)*(2*c^2*x^2 +1))
Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=-\frac {12 \, b^{2} c^{2} x^{2} + 9 \, b^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 9 \, a^{2} + 2 \, b^{2} + 6 \, {\left (3 \, a b - {\left (2 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \, {\left (2 \, a b c^{3} x^{3} + a b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
-1/27*(12*b^2*c^2*x^2 + 9*b^2*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1) /(c*x))^2 + 9*a^2 + 2*b^2 + 6*(3*a*b - (2*b^2*c^3*x^3 + b^2*c*x)*sqrt(-(c^ 2*x^2 - 1)/(c^2*x^2)))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - 6*(2*a*b*c^3*x^3 + a*b*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/x^3
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
2/9*a*b*((c^4*(1/(c^2*x^2) - 1)^(3/2) + 3*c^4*sqrt(1/(c^2*x^2) - 1))/c - 3 *arcsech(c*x)/x^3) + b^2*integrate(log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2/x^4, x) - 1/3*a^2/x^3
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^4} \,d x \]