3.27 \(\int (a+b \text{csch}^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=120 \[ \frac{6 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c}-\frac{6 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,-e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2}{c} \]

[Out]

x*(a + b*ArcCsch[c*x])^3 + (6*b*(a + b*ArcCsch[c*x])^2*ArcTanh[E^ArcCsch[c*x]])/c + (6*b^2*(a + b*ArcCsch[c*x]
)*PolyLog[2, -E^ArcCsch[c*x]])/c - (6*b^2*(a + b*ArcCsch[c*x])*PolyLog[2, E^ArcCsch[c*x]])/c - (6*b^3*PolyLog[
3, -E^ArcCsch[c*x]])/c + (6*b^3*PolyLog[3, E^ArcCsch[c*x]])/c

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Rubi [A]  time = 0.117491, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6280, 5452, 4182, 2531, 2282, 6589} \[ \frac{6 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c}-\frac{6 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,-e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c x)}\right )}{c}+x \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])^3,x]

[Out]

x*(a + b*ArcCsch[c*x])^3 + (6*b*(a + b*ArcCsch[c*x])^2*ArcTanh[E^ArcCsch[c*x]])/c + (6*b^2*(a + b*ArcCsch[c*x]
)*PolyLog[2, -E^ArcCsch[c*x]])/c - (6*b^2*(a + b*ArcCsch[c*x])*PolyLog[2, E^ArcCsch[c*x]])/c - (6*b^3*PolyLog[
3, -E^ArcCsch[c*x]])/c + (6*b^3*PolyLog[3, 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \left (a+b \text{csch}^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \coth (x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{6 b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \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 )^3+\frac{6 b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{6 b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{6 b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{6 b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{Li}_3\left (-e^{\text{csch}^{-1}(c x)}\right )}{c}+\frac{6 b^3 \text{Li}_3\left (e^{\text{csch}^{-1}(c x)}\right )}{c}\\ \end{align*}

Mathematica [B]  time = 0.323685, size = 246, normalized size = 2.05 \[ \frac{3 a b^2 \left (-2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )+2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )+\text{csch}^{-1}(c x) \left (c x \text{csch}^{-1}(c x)-2 \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )+2 \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )\right )\right )}{c}+\frac{b^3 \left (-6 \text{csch}^{-1}(c x) \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )+6 \text{csch}^{-1}(c x) \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )-6 \text{PolyLog}\left (3,-e^{-\text{csch}^{-1}(c x)}\right )+6 \text{PolyLog}\left (3,e^{-\text{csch}^{-1}(c x)}\right )+c x \text{csch}^{-1}(c x)^3-3 \text{csch}^{-1}(c x)^2 \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )+3 \text{csch}^{-1}(c x)^2 \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )\right )}{c}+\frac{3 a^2 b \log \left (c x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{c}+3 a^2 b x \text{csch}^{-1}(c x)+a^3 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsch[c*x])^3,x]

[Out]

a^3*x + 3*a^2*b*x*ArcCsch[c*x] + (3*a^2*b*Log[c*x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/c + (3*a*b^2*(ArcCsch[
c*x]*(c*x*ArcCsch[c*x] - 2*Log[1 - E^(-ArcCsch[c*x])] + 2*Log[1 + E^(-ArcCsch[c*x])]) - 2*PolyLog[2, -E^(-ArcC
sch[c*x])] + 2*PolyLog[2, E^(-ArcCsch[c*x])]))/c + (b^3*(c*x*ArcCsch[c*x]^3 - 3*ArcCsch[c*x]^2*Log[1 - E^(-Arc
Csch[c*x])] + 3*ArcCsch[c*x]^2*Log[1 + E^(-ArcCsch[c*x])] - 6*ArcCsch[c*x]*PolyLog[2, -E^(-ArcCsch[c*x])] + 6*
ArcCsch[c*x]*PolyLog[2, E^(-ArcCsch[c*x])] - 6*PolyLog[3, -E^(-ArcCsch[c*x])] + 6*PolyLog[3, E^(-ArcCsch[c*x])
]))/c

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^3,x)

[Out]

int((a+b*arccsch(c*x))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^3,x, algorithm="maxima")

[Out]

b^3*x*log(sqrt(c^2*x^2 + 1) + 1)^3 + a^3*x + 3/2*(2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - log(sq
rt(1/(c^2*x^2) + 1) - 1))*a^2*b/c - integrate((b^3*log(c)^3 - 3*a*b^2*log(c)^2 + (b^3*c^2*x^2 + b^3)*log(x)^3
+ (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*lo
g(x)^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2 + (b^3*c^2*x^2 + b^3)*log(x) + sqrt(c^2*x^2
+ 1)*(b^3*log(c) - a*b^2 + (b^3*c^2*(log(c) + 1) - a*b^2*c^2)*x^2 + (b^3*c^2*x^2 + b^3)*log(x)))*log(sqrt(c^2*
x^2 + 1) + 1)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2)*log(x) - 3*(
b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2
*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x) + (b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*lo
g(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*
b^2*c^2)*x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + (b^3*log(c)^3 - 3*a*b^2*log(c)^2 + (b^3*
c^2*x^2 + b^3)*log(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log
(c) - a*b^2*c^2)*x^2)*log(x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^
2)*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^2 + (c^2*x^2 + 1)^(3/2) + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} \operatorname{arcsch}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arcsch}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arcsch}\left (c x\right ) + a^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^3,x, algorithm="fricas")

[Out]

integral(b^3*arccsch(c*x)^3 + 3*a*b^2*arccsch(c*x)^2 + 3*a^2*b*arccsch(c*x) + a^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))**3,x)

[Out]

Integral((a + b*acsch(c*x))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^3,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^3, x)