Integrand size = 12, antiderivative size = 126 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=-\frac {3 b \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{2} x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{c^2}+\frac {3 b^3 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \]
-3/2*b*(a+b*arcsech(c*x))^2/c^2+1/2*x^2*(a+b*arcsech(c*x))^3+3*b^2*(a+b*ar csech(c*x))*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)/c^2+3/2*b^3*p olylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)/c^2-3/2*b*(c*x+1)*(a +b*arcsech(c*x))^2*((-c*x+1)/(c*x+1))^(1/2)/c^2
Time = 1.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.74 \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\frac {-3 b^2 \left (-a c^2 x^2+b \left (-1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )\right ) \text {sech}^{-1}(c x)^2+b^3 c^2 x^2 \text {sech}^{-1}(c x)^3+3 b \text {sech}^{-1}(c x) \left (a \left (a c^2 x^2-2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )+2 b^2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+a \left (a \left (a c^2 x^2-3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )+6 b^2 \log \left (\frac {1}{c x}\right )\right )-3 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 c^2} \]
(-3*b^2*(-(a*c^2*x^2) + b*(-1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]))*ArcSech[c*x]^2 + b^3*c^2*x^2*ArcSech[c*x]^3 + 3*b*ArcSec h[c*x]*(a*(a*c^2*x^2 - 2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + 2*b^2*Lo g[1 + E^(-2*ArcSech[c*x])]) + a*(a*(a*c^2*x^2 - 3*b*Sqrt[(1 - c*x)/(1 + c* x)]*(1 + c*x)) + 6*b^2*Log[1/(c*x)]) - 3*b^3*PolyLog[2, -E^(-2*ArcSech[c*x ])])/(2*c^2)
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6839, 5974, 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.
| \(\displaystyle \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 6839 |
| \(\displaystyle -\frac {\int c^2 x^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3d\text {sech}^{-1}(c x)}{c^2}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle -\frac {\frac {3}{2} b \int c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2d\text {sech}^{-1}(c x)-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3}{c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \csc \left (i \text {sech}^{-1}(c x)+\frac {\pi }{2}\right )^2d\text {sech}^{-1}(c x)}{c^2}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2-2 i b \int -i \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )}{c^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2-2 b \int \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3}{c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2-2 b \int -i \left (a+b \text {sech}^{-1}(c x)\right ) \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )}{c^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2+2 i b \int \left (a+b \text {sech}^{-1}(c x)\right ) \tan \left (i \text {sech}^{-1}(c x)\right )d\text {sech}^{-1}(c x)\right )}{c^2}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2+2 i b \left (2 i \int \frac {e^{2 \text {sech}^{-1}(c x)} \left (a+b \text {sech}^{-1}(c x)\right )}{1+e^{2 \text {sech}^{-1}(c x)}}d\text {sech}^{-1}(c x)-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}\right )\right )}{c^2}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2+2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{2} b \int \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )d\text {sech}^{-1}(c x)\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}\right )\right )}{c^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2+2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{4} b \int e^{-2 \text {sech}^{-1}(c x)} \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )de^{2 \text {sech}^{-1}(c x)}\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}\right )\right )}{c^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {-\frac {1}{2} c^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3}{2} b \left (\sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2+2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )\right )-\frac {i \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}\right )\right )}{c^2}\) |
-((-1/2*(c^2*x^2*(a + b*ArcSech[c*x])^3) + (3*b*(Sqrt[(1 - c*x)/(1 + c*x)] *(1 + c*x)*(a + b*ArcSech[c*x])^2 + (2*I)*b*(((-1/2*I)*(a + b*ArcSech[c*x] )^2)/b + (2*I)*(((a + b*ArcSech[c*x])*Log[1 + E^(2*ArcSech[c*x])])/2 + (b* PolyLog[2, -E^(2*ArcSech[c*x])])/4))))/2)/c^2)
3.1.44.3.1 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[((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])
Time = 0.78 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.52
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} c^{2} x^{2}}{2}+b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-3 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3\right )}{2}-3 \operatorname {arcsech}\left (c x \right )^{2}+3 \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )+3 a \,b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )+3 b \,a^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(318\) |
| default | \(\frac {\frac {a^{3} c^{2} x^{2}}{2}+b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-3 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3\right )}{2}-3 \operatorname {arcsech}\left (c x \right )^{2}+3 \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )+3 a \,b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )+3 b \,a^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(318\) |
| parts | \(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (\frac {\operatorname {arcsech}\left (c x \right )^{2} \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-3 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3\right )}{2}-3 \operatorname {arcsech}\left (c x \right )^{2}+3 \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )}{c^{2}}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsech}\left (c x \right )+\frac {\operatorname {arcsech}\left (c x \right ) \left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+2\right )}{2}+\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\right )}{c^{2}}+\frac {3 b \,a^{2} \left (\frac {c^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\) | \(320\) |
1/c^2*(1/2*a^3*c^2*x^2+b^3*(1/2*arcsech(c*x)^2*(c^2*x^2*arcsech(c*x)-3*(-( c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+3)-3*arcsech(c*x)^2+3*arcsech(c* x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)+3/2*polylog(2,-(1/c/x+ (-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2))+3*a*b^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))+3*b*a^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)))
\[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x \,d x } \]
integral(b^3*x*arcsech(c*x)^3 + 3*a*b^2*x*arcsech(c*x)^2 + 3*a^2*b*x*arcse ch(c*x) + a^3*x, x)
\[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int x \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \]
\[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x \,d x } \]
3/2*a*b^2*x^2*arcsech(c*x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arcsech(c*x) - x*sqr t(1/(c^2*x^2) - 1)/c)*a^2*b - 3*(x*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x)/c + log(x)/c^2)*a*b^2 + b^3*integrate(x*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1 ) + 1/(c*x))^3, x)
\[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x \,d x } \]
Timed out. \[ \int x \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]