Integrand size = 6, antiderivative size = 63 \[ \int \text {sech}^{-1}(a x)^2 \, dx=x \text {sech}^{-1}(a x)^2-\frac {4 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )}{a}+\frac {2 i \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{a}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{a} \]
x*arcsech(a*x)^2-4*arcsech(a*x)*arctan(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/ 2))/a+2*I*polylog(2,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a-2*I*poly log(2,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43 \[ \int \text {sech}^{-1}(a x)^2 \, dx=\frac {i \left (\text {sech}^{-1}(a x) \left (-i a x \text {sech}^{-1}(a x)+2 \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )\right )}{a} \]
(I*(ArcSech[a*x]*((-I)*a*x*ArcSech[a*x] + 2*Log[1 - I/E^ArcSech[a*x]] - 2* Log[1 + I/E^ArcSech[a*x]]) + 2*PolyLog[2, (-I)/E^ArcSech[a*x]] - 2*PolyLog [2, I/E^ArcSech[a*x]]))/a
Time = 0.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6833, 5941, 3042, 4668, 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.
| \(\displaystyle \int \text {sech}^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6833 |
| \(\displaystyle -\frac {\int a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)}{a}\) |
| \(\Big \downarrow \) 5941 |
| \(\displaystyle -\frac {2 \int a x \text {sech}^{-1}(a x)d\text {sech}^{-1}(a x)-a x \text {sech}^{-1}(a x)^2}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-a x \text {sech}^{-1}(a x)^2+2 \int \text {sech}^{-1}(a x) \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)}{a}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {-a x \text {sech}^{-1}(a x)^2+2 \left (-i \int \log \left (1-i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+i \int \log \left (1+i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {-a x \text {sech}^{-1}(a x)^2+2 \left (-i \int e^{-\text {sech}^{-1}(a x)} \log \left (1-i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}+i \int e^{-\text {sech}^{-1}(a x)} \log \left (1+i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}+2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {-a x \text {sech}^{-1}(a x)^2+2 \left (2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )\right )}{a}\) |
-((-(a*x*ArcSech[a*x]^2) + 2*(2*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]] - I*Po lyLog[2, (-I)*E^ArcSech[a*x]] + I*PolyLog[2, I*E^ArcSech[a*x]]))/a)
3.1.5.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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_Symbol] :> Simp[-c^(-1) S ubst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
Time = 0.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.90
| method | result | size |
| derivativedivides | \(\frac {\operatorname {arcsech}\left (a x \right )^{2} a x +2 i \operatorname {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )-2 i \operatorname {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a}\) | \(183\) |
| default | \(\frac {\operatorname {arcsech}\left (a x \right )^{2} a x +2 i \operatorname {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )-2 i \operatorname {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{a}\) | \(183\) |
1/a*(arcsech(a*x)^2*a*x+2*I*arcsech(a*x)*ln(1+I*(1/a/x+(1/a/x-1)^(1/2)*(1+ 1/a/x)^(1/2)))-2*I*arcsech(a*x)*ln(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1 /2)))+2*I*dilog(1+I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))-2*I*dilog(1-I *(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))))
\[ \int \text {sech}^{-1}(a x)^2 \, dx=\int { \operatorname {arsech}\left (a x\right )^{2} \,d x } \]
\[ \int \text {sech}^{-1}(a x)^2 \, dx=\int \operatorname {asech}^{2}{\left (a x \right )}\, dx \]
\[ \int \text {sech}^{-1}(a x)^2 \, dx=\int { \operatorname {arsech}\left (a x\right )^{2} \,d x } \]
x*log(sqrt(a*x + 1)*sqrt(-a*x + 1) + 1)^2 - integrate(-(a^2*x^2*log(a)^2 + (a^2*x^2 - 1)*log(x)^2 + (a^2*x^2*log(a)^2 + (a^2*x^2 - 1)*log(x)^2 - log (a)^2 + 2*(a^2*x^2*log(a) - log(a))*log(x))*sqrt(a*x + 1)*sqrt(-a*x + 1) - 2*(a^2*x^2*log(a) + (a^2*x^2*(log(a) + 1) + (a^2*x^2 - 1)*log(x) - log(a) )*sqrt(a*x + 1)*sqrt(-a*x + 1) + (a^2*x^2 - 1)*log(x) - log(a))*log(sqrt(a *x + 1)*sqrt(-a*x + 1) + 1) - log(a)^2 + 2*(a^2*x^2*log(a) - log(a))*log(x ))/(a^2*x^2 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(-a*x + 1) - 1), x)
\[ \int \text {sech}^{-1}(a x)^2 \, dx=\int { \operatorname {arsech}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int \text {sech}^{-1}(a x)^2 \, dx=\int {\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2 \,d x \]