Integrand size = 12, antiderivative size = 65 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{c^2} \]
1/2*x^2*(a+b*arcsech(c*x))^2-b^2*ln(x)/c^2-b*(c*x+1)*(a+b*arcsech(c*x))*(( -c*x+1)/(c*x+1))^(1/2)/c^2
Time = 0.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.72 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\frac {a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )-2 b \left (-a c^2 x^2+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)+b^2 c^2 x^2 \text {sech}^{-1}(c x)^2-2 b^2 \log (c x)}{2 c^2} \]
(a*(a*c^2*x^2 - 2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) - 2*b*(-(a*c^2*x^ 2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + b^2*c^2*x^2*Arc Sech[c*x]^2 - 2*b^2*Log[c*x])/(2*c^2)
Time = 0.41 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6839, 5974, 3042, 4672, 26, 3042, 26, 3956}
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.
\(\displaystyle \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx\) |
\(\Big \downarrow \) 6839 |
\(\displaystyle -\frac {\int c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 5974 |
\(\displaystyle -\frac {b \int c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2+b \int \left (a+b \text {sech}^{-1}(c x)\right ) \csc \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^2d\text {sech}^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2+b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-i b \int -i \sqrt {\frac {1-c x}{c x+1}} (c x+1)d\text {sech}^{-1}(c x)\right )}{c^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \int \sqrt {\frac {1-c x}{c x+1}} (c x+1)d\text {sech}^{-1}(c x)\right )-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2+b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \int -i \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )}{c^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2+b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )+i b \int \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )}{c^2}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )-b \log \left (\frac {1}{c x}\right )\right )-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^2}\) |
-((-1/2*(c^2*x^2*(a + b*ArcSech[c*x])^2) + b*(Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]) - b*Log[1/(c*x)]))/c^2)
3.1.35.3.1 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[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && 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. \(164\) vs. \(2(61)=122\).
Time = 0.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.54
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(165\) |
derivativedivides | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}+b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(166\) |
default | \(\frac {\frac {a^{2} c^{2} x^{2}}{2}+b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(166\) |
1/2*a^2*x^2+b^2/c^2*(-2*arcsech(c*x)+1/2*arcsech(c*x)*(c^2*x^2*arcsech(c*x )-2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+2)+ln(1+(1/c/x+(-1+1/c/x) ^(1/2)*(1+1/c/x)^(1/2))^2))+2*a*b/c^2*(1/2*c^2*x^2*arcsech(c*x)-1/2*(-(c*x -1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (61) = 122\).
Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.15 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\frac {b^{2} c^{2} x^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + a^{2} c^{2} x^{2} - 2 \, a b c^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, a b c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, b^{2} \log \left (x\right ) + 2 \, {\left (a b c^{2} x^{2} - b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b c^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c^{2}} \]
1/2*(b^2*c^2*x^2*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^2 + a ^2*c^2*x^2 - 2*a*b*c^2*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) - 2 *a*b*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*b^2*log(x) + 2*(a*b*c^2*x^2 - b^2*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - a*b*c^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/c^2
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.52 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {asech}{\left (c x \right )} - \frac {a b \sqrt {- c^{2} x^{2} + 1}}{c^{2}} + \frac {b^{2} x^{2} \operatorname {asech}^{2}{\left (c x \right )}}{2} - \frac {b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asech}{\left (c x \right )}}{c^{2}} - \frac {b^{2} \log {\left (x \right )}}{c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \infty b\right )^{2}}{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*x**2/2 + a*b*x**2*asech(c*x) - a*b*sqrt(-c**2*x**2 + 1)/c* *2 + b**2*x**2*asech(c*x)**2/2 - b**2*sqrt(-c**2*x**2 + 1)*asech(c*x)/c**2 - b**2*log(x)/c**2, Ne(c, 0)), (x**2*(a + oo*b)**2/2, True))
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} a b - {\left (\frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )}{c} + \frac {\log \left (x\right )}{c^{2}}\right )} b^{2} \]
1/2*b^2*x^2*arcsech(c*x)^2 + 1/2*a^2*x^2 + (x^2*arcsech(c*x) - x*sqrt(1/(c ^2*x^2) - 1)/c)*a*b - (x*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x)/c + log(x)/c^2 )*b^2
\[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x \,d x } \]
Timed out. \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]