3.5 \(\int \text{sech}^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2 i \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a x)}\right )}{a}-\frac{2 i \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a x)}\right )}{a}+x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a} \]

[Out]

x*ArcSech[a*x]^2 - (4*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]])/a + ((2*I)*PolyLog[2, (-I)*E^ArcSech[a*x]])/a - ((2
*I)*PolyLog[2, I*E^ArcSech[a*x]])/a

________________________________________________________________________________________

Rubi [A]  time = 0.0550174, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6279, 5418, 4180, 2279, 2391} \[ \frac{2 i \text{PolyLog}\left (2,-i e^{\text{sech}^{-1}(a x)}\right )}{a}-\frac{2 i \text{PolyLog}\left (2,i e^{\text{sech}^{-1}(a x)}\right )}{a}+x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[a*x]^2,x]

[Out]

x*ArcSech[a*x]^2 - (4*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]])/a + ((2*I)*PolyLog[2, (-I)*E^ArcSech[a*x]])/a - ((2
*I)*PolyLog[2, I*E^ArcSech[a*x]])/a

Rule 6279

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[c^(-1), Subst[Int[(a + b*x)^n*Sech[x]*Tanh[x]
, x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]

Rule 5418

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] + Dist[(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]

Rule 4180

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] + (-Dist[(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] + Dist[(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]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2391

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]

Rubi steps

\begin{align*} \int \text{sech}^{-1}(a x)^2 \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \text{sech}(x) \tanh (x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a}\\ &=x \text{sech}^{-1}(a x)^2-\frac{2 \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a}\\ &=x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a}+\frac{(2 i) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text{sech}^{-1}(a x)\right )}{a}-\frac{(2 i) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text{sech}^{-1}(a x)\right )}{a}\\ &=x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a x)}\right )}{a}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\text{sech}^{-1}(a x)}\right )}{a}\\ &=x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left (e^{\text{sech}^{-1}(a x)}\right )}{a}+\frac{2 i \text{Li}_2\left (-i e^{\text{sech}^{-1}(a x)}\right )}{a}-\frac{2 i \text{Li}_2\left (i e^{\text{sech}^{-1}(a x)}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.185302, size = 90, normalized size = 1.43 \[ \frac{i \left (2 \text{PolyLog}\left (2,-i e^{-\text{sech}^{-1}(a x)}\right )-2 \text{PolyLog}\left (2,i e^{-\text{sech}^{-1}(a x)}\right )+\text{sech}^{-1}(a x) \left (-i a x \text{sech}^{-1}(a x)+2 \log \left (1-i e^{-\text{sech}^{-1}(a x)}\right )-2 \log \left (1+i e^{-\text{sech}^{-1}(a x)}\right )\right )\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSech[a*x]^2,x]

[Out]

(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*PolyL
og[2, (-I)/E^ArcSech[a*x]] - 2*PolyLog[2, I/E^ArcSech[a*x]]))/a

________________________________________________________________________________________

Maple [A]  time = 0.272, size = 190, normalized size = 3. \begin{align*} x \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}+{\frac{2\,i{\rm arcsech} \left (ax\right )}{a}\ln \left ( 1+i \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) \right ) }-{\frac{2\,i{\rm arcsech} \left (ax\right )}{a}\ln \left ( 1-i \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) \right ) }-{\frac{2\,i}{a}{\it dilog} \left ( 1-i \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) \right ) }+{\frac{2\,i}{a}{\it dilog} \left ( 1+i \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(a*x)^2,x)

[Out]

x*arcsech(a*x)^2+2*I/a*arcsech(a*x)*ln(1+I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))-2*I/a*arcsech(a*x)*ln(1-I*
(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))-2*I/a*dilog(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))+2*I/a*dilog(
1+I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} x \log \left (\sqrt{a x + 1} \sqrt{-a x + 1} + 1\right )^{2} - \int -\frac{a^{2} x^{2} \log \left (a\right )^{2} +{\left (a^{2} x^{2} - 1\right )} \log \left (x\right )^{2} +{\left (a^{2} x^{2} \log \left (a\right )^{2} +{\left (a^{2} x^{2} - 1\right )} \log \left (x\right )^{2} - \log \left (a\right )^{2} + 2 \,{\left (a^{2} x^{2} \log \left (a\right ) - \log \left (a\right )\right )} \log \left (x\right )\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 2 \,{\left (a^{2} x^{2} \log \left (a\right ) +{\left (a^{2} x^{2}{\left (\log \left (a\right ) + 1\right )} +{\left (a^{2} x^{2} - 1\right )} \log \left (x\right ) - \log \left (a\right )\right )} \sqrt{a x + 1} \sqrt{-a x + 1} +{\left (a^{2} x^{2} - 1\right )} \log \left (x\right ) - \log \left (a\right )\right )} \log \left (\sqrt{a x + 1} \sqrt{-a x + 1} + 1\right ) - \log \left (a\right )^{2} + 2 \,{\left (a^{2} x^{2} \log \left (a\right ) - \log \left (a\right )\right )} \log \left (x\right )}{a^{2} x^{2} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2,x, algorithm="maxima")

[Out]

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*l
og(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))*l
og(x))/(a^2*x^2 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(-a*x + 1) - 1), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{arsech}\left (a x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2,x, algorithm="fricas")

[Out]

integral(arcsech(a*x)^2, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{asech}^{2}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(a*x)**2,x)

[Out]

Integral(asech(a*x)**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arsech}\left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(a*x)^2,x, algorithm="giac")

[Out]

integrate(arcsech(a*x)^2, x)