Integrand size = 14, antiderivative size = 61 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=-\frac {2 b^2}{x}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x} \]
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=-\frac {a^2+2 b^2-2 a b \sqrt {\frac {1-c x}{1+c x}} (1+c x)-2 b \left (-a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 \text {sech}^{-1}(c x)^2}{x} \]
-((a^2 + 2*b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 2*b*(-a + b*S qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + b^2*ArcSech[c*x]^2)/x)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6839, 3042, 26, 3777, 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.
\(\displaystyle \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx\) |
\(\Big \downarrow \) 6839 |
\(\displaystyle -c \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \int -i \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i c \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \int \frac {a+b \text {sech}^{-1}(c x)}{c x}d\text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \int \left (a+b \text {sech}^{-1}(c x)\right ) \sin \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-i b \int -\frac {i \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}d\text {sech}^{-1}(c x)\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-b \int \frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}d\text {sech}^{-1}(c x)\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-b \int -i \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}+i b \int \sin \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle i c \left (\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{c x}-2 i b \left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{c x}-\frac {b}{c x}\right )\right )\) |
I*c*((I*(a + b*ArcSech[c*x])^2)/(c*x) - (2*I)*b*(-(b/(c*x)) + (Sqrt[(1 - c *x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(c*x)))
3.1.38.3.1 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[((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])
Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs. \(2(59)=118\).
Time = 0.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.98
method | result | size |
parts | \(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+2 a b c \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\) | \(121\) |
derivativedivides | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(124\) |
default | \(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+2 a b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\) | \(124\) |
-a^2/x+b^2*c*(-1/c/x*arcsech(c*x)^2+2*arcsech(c*x)*(-(c*x-1)/c/x)^(1/2)*(( c*x+1)/c/x)^(1/2)-2/c/x)+2*a*b*c*(-1/c/x*arcsech(c*x)+(-(c*x-1)/c/x)^(1/2) *((c*x+1)/c/x)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (59) = 118\).
Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=\frac {2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - b^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - a^{2} - 2 \, b^{2} + 2 \, {\left (b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \]
(2*a*b*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - b^2*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 - a^2 - 2*b^2 + 2*(b^2*c*x*sqrt(-(c^2*x^2 - 1) /(c^2*x^2)) - a*b)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/x
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} a b + 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right ) - \frac {1}{x}\right )} b^{2} - \frac {b^{2} \operatorname {arsech}\left (c x\right )^{2}}{x} - \frac {a^{2}}{x} \]
2*(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*a*b + 2*(c*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x) - 1/x)*b^2 - b^2*arcsech(c*x)^2/x - a^2/x
\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^2} \,d x \]