Integrand size = 8, antiderivative size = 102 \[ \int x \text {sech}^{-1}(a x)^3 \, dx=-\frac {3 \text {sech}^{-1}(a x)^2}{2 a^2}-\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a x)^3+\frac {3 \text {sech}^{-1}(a x) \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )}{a^2}+\frac {3 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )}{2 a^2} \]
-3/2*arcsech(a*x)^2/a^2+1/2*x^2*arcsech(a*x)^3+3*arcsech(a*x)*ln(1+(1/a/x+ (1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))^2)/a^2+3/2*polylog(2,-(1/a/x+(1/a/x-1)^(1 /2)*(1+1/a/x)^(1/2))^2)/a^2-3/2*(a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^ (1/2)/a^2
Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99 \[ \int x \text {sech}^{-1}(a x)^3 \, dx=\frac {\text {sech}^{-1}(a x) \left (-3 \left (-1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right ) \text {sech}^{-1}(a x)+a^2 x^2 \text {sech}^{-1}(a x)^2+6 \log \left (1+e^{-2 \text {sech}^{-1}(a x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a x)}\right )}{2 a^2} \]
(ArcSech[a*x]*(-3*(-1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)])*ArcSech[a*x] + a^2*x^2*ArcSech[a*x]^2 + 6*Log[1 + E^(-2*ArcSech[a *x])]) - 3*PolyLog[2, -E^(-2*ArcSech[a*x])])/(2*a^2)
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {6839, 5941, 3042, 4672, 26, 3042, 26, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int x \text {sech}^{-1}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -\frac {\int a^2 x^2 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^3d\text {sech}^{-1}(a x)}{a^2}\) |
| \(\Big \downarrow \) 5941 |
| \(\displaystyle -\frac {\frac {3}{2} \int a^2 x^2 \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^2d\text {sech}^{-1}(a x)}{a^2}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2-2 i \int -i \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)d\text {sech}^{-1}(a x)\right )}{a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2-2 \int \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)d\text {sech}^{-1}(a x)\right )-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2-2 \int -i \text {sech}^{-1}(a x) \tan \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\right )}{a^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+2 i \int \text {sech}^{-1}(a x) \tan \left (i \text {sech}^{-1}(a x)\right )d\text {sech}^{-1}(a x)\right )}{a^2}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+2 i \left (2 i \int \frac {e^{2 \text {sech}^{-1}(a x)} \text {sech}^{-1}(a x)}{1+e^{2 \text {sech}^{-1}(a x)}}d\text {sech}^{-1}(a x)-\frac {1}{2} i \text {sech}^{-1}(a x)^2\right )\right )}{a^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+2 i \left (2 i \left (\frac {1}{2} \text {sech}^{-1}(a x) \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)\right )-\frac {1}{2} i \text {sech}^{-1}(a x)^2\right )\right )}{a^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+2 i \left (2 i \left (\frac {1}{2} \text {sech}^{-1}(a x) \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right )-\frac {1}{4} \int e^{-2 \text {sech}^{-1}(a x)} \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )de^{2 \text {sech}^{-1}(a x)}\right )-\frac {1}{2} i \text {sech}^{-1}(a x)^2\right )\right )}{a^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {-\frac {1}{2} a^2 x^2 \text {sech}^{-1}(a x)^3+\frac {3}{2} \left (\sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+2 i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \text {sech}^{-1}(a x) \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right )\right )-\frac {1}{2} i \text {sech}^{-1}(a x)^2\right )\right )}{a^2}\) |
-((-1/2*(a^2*x^2*ArcSech[a*x]^3) + (3*(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) *ArcSech[a*x]^2 + (2*I)*((-1/2*I)*ArcSech[a*x]^2 + (2*I)*((ArcSech[a*x]*Lo g[1 + E^(2*ArcSech[a*x])])/2 + PolyLog[2, -E^(2*ArcSech[a*x])]/4))))/2)/a^ 2)
3.1.13.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
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[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_) ^(n_.)]^(q_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p )), x] + Simp[(m - n + 1)/(b*n*p) Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 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 = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arcsech}\left (a x \right )^{2} \left (-3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x +\operatorname {arcsech}\left (a x \right ) a^{2} x^{2}+3\right )}{2}-3 \operatorname {arcsech}\left (a x \right )^{2}+3 \,\operatorname {arcsech}\left (a x \right ) \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}}{a^{2}}\) | \(149\) |
| default | \(\frac {\frac {\operatorname {arcsech}\left (a x \right )^{2} \left (-3 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x +\operatorname {arcsech}\left (a x \right ) a^{2} x^{2}+3\right )}{2}-3 \operatorname {arcsech}\left (a x \right )^{2}+3 \,\operatorname {arcsech}\left (a x \right ) \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2}}{a^{2}}\) | \(149\) |
1/a^2*(1/2*arcsech(a*x)^2*(-3*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*a*x +arcsech(a*x)*a^2*x^2+3)-3*arcsech(a*x)^2+3*arcsech(a*x)*ln(1+(1/a/x+(1/a/ x-1)^(1/2)*(1+1/a/x)^(1/2))^2)+3/2*polylog(2,-(1/a/x+(1/a/x-1)^(1/2)*(1+1/ a/x)^(1/2))^2))
\[ \int x \text {sech}^{-1}(a x)^3 \, dx=\int { x \operatorname {arsech}\left (a x\right )^{3} \,d x } \]
\[ \int x \text {sech}^{-1}(a x)^3 \, dx=\int x \operatorname {asech}^{3}{\left (a x \right )}\, dx \]
\[ \int x \text {sech}^{-1}(a x)^3 \, dx=\int { x \operatorname {arsech}\left (a x\right )^{3} \,d x } \]
\[ \int x \text {sech}^{-1}(a x)^3 \, dx=\int { x \operatorname {arsech}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x \text {sech}^{-1}(a x)^3 \, dx=\int x\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3 \,d x \]