### 3.4 $$\int \frac{\text{csch}^{-1}(a+b x)}{x} \, dx$$

Optimal. Leaf size=162 $\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )$

[Out]

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

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Rubi [A]  time = 0.286667, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.8, Rules used = {6322, 5596, 5569, 3716, 2190, 2279, 2391, 5561} $\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[ArcCsch[a + b*x]/x,x]

[Out]

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

Rule 6322

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Dist[(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ
[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]

Rule 5596

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + (d_.)*(x_)]^(p_.))/(Csch[(c_.) + (
d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n*G[c + d*x]^p)/(b + a*Sinh[c
+ d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]

Rule 5569

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[c +
d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

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]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin{align*} \int \frac{\text{csch}^{-1}(a+b x)}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{x \coth (x) \text{csch}(x)}{-a+\text{csch}(x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x \coth (x)}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=-\left (a \operatorname{Subst}\left (\int \frac{x \cosh (x)}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\right )-\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^x x}{1-\sqrt{1+a^2}-a e^x} \, dx,x,\text{csch}^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^x x}{1+\sqrt{1+a^2}-a e^x} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \log \left (1-\frac{a e^x}{1-\sqrt{1+a^2}}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )-\operatorname{Subst}\left (\int \log \left (1-\frac{a e^x}{1+\sqrt{1+a^2}}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )+\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{1-\sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{1+\sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(a+b x)}\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{Li}_2\left (\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{Li}_2\left (\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\frac{1}{2} \text{Li}_2\left (e^{2 \text{csch}^{-1}(a+b x)}\right )\\ \end{align*}

Mathematica [C]  time = 0.378753, size = 427, normalized size = 2.64 $\frac{1}{8} \left (8 \text{PolyLog}\left (2,\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+8 \text{PolyLog}\left (2,-\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+4 \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(a+b x)}\right )+8 \text{csch}^{-1}(a+b x) \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+8 \text{csch}^{-1}(a+b x) \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )+4 i \pi \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+4 i \pi \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )+16 i \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )-16 i \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )-32 \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{(1-i a) \cot \left (\frac{1}{4} \left (\pi +2 i \text{csch}^{-1}(a+b x)\right )\right )}{\sqrt{a^2+1}}\right )-4 i \pi \log \left (\frac{b x}{a+b x}\right )-8 \text{csch}^{-1}(a+b x)^2-4 i \pi \text{csch}^{-1}(a+b x)-8 \text{csch}^{-1}(a+b x) \log \left (1-e^{-2 \text{csch}^{-1}(a+b x)}\right )+\pi ^2\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCsch[a + b*x]/x,x]

[Out]

(Pi^2 - (4*I)*Pi*ArcCsch[a + b*x] - 8*ArcCsch[a + b*x]^2 - 32*ArcSin[Sqrt[(-I + a)/a]/Sqrt[2]]*ArcTan[((1 - I*
a)*Cot[(Pi + (2*I)*ArcCsch[a + b*x])/4])/Sqrt[1 + a^2]] - 8*ArcCsch[a + b*x]*Log[1 - E^(-2*ArcCsch[a + b*x])]
+ (4*I)*Pi*Log[1 - ((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + 8*ArcCsch[a + b*x]*Log[1 - ((-1 + Sqrt[1 + a
^2])*E^ArcCsch[a + b*x])/a] + (16*I)*ArcSin[Sqrt[(-I + a)/a]/Sqrt[2]]*Log[1 - ((-1 + Sqrt[1 + a^2])*E^ArcCsch[
a + b*x])/a] + (4*I)*Pi*Log[1 + ((1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + 8*ArcCsch[a + b*x]*Log[1 + ((1 +
Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] - (16*I)*ArcSin[Sqrt[(-I + a)/a]/Sqrt[2]]*Log[1 + ((1 + Sqrt[1 + a^2])*
E^ArcCsch[a + b*x])/a] - (4*I)*Pi*Log[(b*x)/(a + b*x)] + 4*PolyLog[2, E^(-2*ArcCsch[a + b*x])] + 8*PolyLog[2,
((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + 8*PolyLog[2, -(((1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a)])/8

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Maple [F]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccsch} \left (bx+a\right )}{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(arccsch(b*x+a)/x,x)

[Out]

int(arccsch(b*x+a)/x,x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(b*x+a)/x,x, algorithm="maxima")

[Out]

integrate(arccsch(b*x + a)/x, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(b*x+a)/x,x, algorithm="fricas")

[Out]

integral(arccsch(b*x + a)/x, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsch(b*x+a)/x,x)

[Out]

Integral(acsch(a + b*x)/x, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(b*x+a)/x,x, algorithm="giac")

[Out]

integrate(arccsch(b*x + a)/x, x)