3.18 \(\int (a+b \text{csch}^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=68 \[ \frac{2 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{2 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{4 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c} \]

[Out]

x*(a + b*ArcCsch[c*x])^2 + (4*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/c + (2*b^2*PolyLog[2, -E^ArcCsch
[c*x]])/c - (2*b^2*PolyLog[2, E^ArcCsch[c*x]])/c

________________________________________________________________________________________

Rubi [A]  time = 0.0676694, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6280, 5452, 4182, 2279, 2391} \[ \frac{2 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{2 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{4 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])^2,x]

[Out]

x*(a + b*ArcCsch[c*x])^2 + (4*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/c + (2*b^2*PolyLog[2, -E^ArcCsch
[c*x]])/c - (2*b^2*PolyLog[2, E^ArcCsch[c*x]])/c

Rule 6280

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

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && 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 \left (a+b \text{csch}^{-1}(c x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{4 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{4 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{4 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{2 b^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{2 b^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c}\\ \end{align*}

Mathematica [A]  time = 0.222216, size = 121, normalized size = 1.78 \[ \frac{-2 b^2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )+2 b^2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )+a^2 c x+2 a b c x \text{csch}^{-1}(c x)-2 a b \log \left (\tanh \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )\right )+b^2 c x \text{csch}^{-1}(c x)^2-2 b^2 \text{csch}^{-1}(c x) \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )+2 b^2 \text{csch}^{-1}(c x) \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )}{c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsch[c*x])^2,x]

[Out]

(a^2*c*x + 2*a*b*c*x*ArcCsch[c*x] + b^2*c*x*ArcCsch[c*x]^2 - 2*b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] + 2
*b^2*ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])] - 2*a*b*Log[Tanh[ArcCsch[c*x]/2]] - 2*b^2*PolyLog[2, -E^(-ArcCsch
[c*x])] + 2*b^2*PolyLog[2, E^(-ArcCsch[c*x])])/c

________________________________________________________________________________________

Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^2,x)

[Out]

int((a+b*arccsch(c*x))^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (x \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )^{2} - \int -\frac{c^{2} x^{2} \log \left (c\right )^{2} +{\left (c^{2} x^{2} + 1\right )} \log \left (x\right )^{2} + \log \left (c\right )^{2} + 2 \,{\left (c^{2} x^{2} \log \left (c\right ) + \log \left (c\right )\right )} \log \left (x\right ) - 2 \,{\left (c^{2} x^{2} \log \left (c\right ) +{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) +{\left (c^{2} x^{2}{\left (\log \left (c\right ) + 1\right )} +{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) + \log \left (c\right )\right )} \sqrt{c^{2} x^{2} + 1} + \log \left (c\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) +{\left (c^{2} x^{2} \log \left (c\right )^{2} +{\left (c^{2} x^{2} + 1\right )} \log \left (x\right )^{2} + \log \left (c\right )^{2} + 2 \,{\left (c^{2} x^{2} \log \left (c\right ) + \log \left (c\right )\right )} \log \left (x\right )\right )} \sqrt{c^{2} x^{2} + 1}}{c^{2} x^{2} +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 1}\,{d x}\right )} b^{2} + a^{2} x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} a b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2,x, algorithm="maxima")

[Out]

(x*log(sqrt(c^2*x^2 + 1) + 1)^2 - integrate(-(c^2*x^2*log(c)^2 + (c^2*x^2 + 1)*log(x)^2 + log(c)^2 + 2*(c^2*x^
2*log(c) + log(c))*log(x) - 2*(c^2*x^2*log(c) + (c^2*x^2 + 1)*log(x) + (c^2*x^2*(log(c) + 1) + (c^2*x^2 + 1)*l
og(x) + log(c))*sqrt(c^2*x^2 + 1) + log(c))*log(sqrt(c^2*x^2 + 1) + 1) + (c^2*x^2*log(c)^2 + (c^2*x^2 + 1)*log
(x)^2 + log(c)^2 + 2*(c^2*x^2*log(c) + log(c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^2 + (c^2*x^2 + 1)^(3/2) + 1),
x))*b^2 + a^2*x + (2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - log(sqrt(1/(c^2*x^2) + 1) - 1))*a*b/
c

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2,x, algorithm="fricas")

[Out]

integral(b^2*arccsch(c*x)^2 + 2*a*b*arccsch(c*x) + a^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))**2,x)

[Out]

Integral((a + b*acsch(c*x))**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^2, x)