3.331 \(\int \frac{\sinh ^{-1}(a x)^3}{c+a^2 c x^2} \, dx\)

Optimal. Leaf size=174 \[ -\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \text{PolyLog}\left (4,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c} \]

[Out]

(2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(a*c) - ((3*I)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c)
+ ((3*I)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c) + ((6*I)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a
*x]])/(a*c) - ((6*I)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c) - ((6*I)*PolyLog[4, (-I)*E^ArcSinh[a*x]]
)/(a*c) + ((6*I)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c)

________________________________________________________________________________________

Rubi [A]  time = 0.130053, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5693, 4180, 2531, 6609, 2282, 6589} \[ -\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \text{PolyLog}\left (4,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]

[Out]

(2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(a*c) - ((3*I)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c)
+ ((3*I)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c) + ((6*I)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a
*x]])/(a*c) - ((6*I)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c) - ((6*I)*PolyLog[4, (-I)*E^ArcSinh[a*x]]
)/(a*c) + ((6*I)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c)

Rule 5693

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

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 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 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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 \frac{\sinh ^{-1}(a x)^3}{c+a^2 c x^2} \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{(3 i) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}+\frac{(3 i) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{(6 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}-\frac{(6 i) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{(6 i) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}+\frac{(6 i) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{(6 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{(6 i) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c}\\ &=\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac{6 i \text{Li}_4\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac{6 i \text{Li}_4\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}\\ \end{align*}

Mathematica [B]  time = 0.219007, size = 454, normalized size = 2.61 \[ -\frac{i \left (192 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \sinh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(a x)}\right )-384 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )-48 \left (\pi -2 i \sinh ^{-1}(a x)\right )^2 \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(a x)}\right )-192 i \pi \text{PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{-\sinh ^{-1}(a x)}\right )+384 \text{PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )-16 \sinh ^{-1}(a x)^4-32 i \pi \sinh ^{-1}(a x)^3+24 \pi ^2 \sinh ^{-1}(a x)^2+8 i \pi ^3 \sinh ^{-1}(a x)-64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )-96 i \pi \sinh ^{-1}(a x)^2 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+96 i \pi \sinh ^{-1}(a x)^2 \log \left (1-i e^{\sinh ^{-1}(a x)}\right )+48 \pi ^2 \sinh ^{-1}(a x) \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \sinh ^{-1}(a x) \log \left (1-i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \sinh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{64 a c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]

[Out]

((-I/64)*(7*Pi^4 + (8*I)*Pi^3*ArcSinh[a*x] + 24*Pi^2*ArcSinh[a*x]^2 - (32*I)*Pi*ArcSinh[a*x]^3 - 16*ArcSinh[a*
x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcSinh[a*x]] + 48*Pi^2*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] - (96*I)*Pi*ArcSi
nh[a*x]^2*Log[1 + I/E^ArcSinh[a*x]] - 64*ArcSinh[a*x]^3*Log[1 + I/E^ArcSinh[a*x]] - 48*Pi^2*ArcSinh[a*x]*Log[1
- I*E^ArcSinh[a*x]] + (96*I)*Pi*ArcSinh[a*x]^2*Log[1 - I*E^ArcSinh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcSinh[a*x
]] + 64*ArcSinh[a*x]^3*Log[1 + I*E^ArcSinh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcSinh[a*x])/4]] - 48*(Pi -
(2*I)*ArcSinh[a*x])^2*PolyLog[2, (-I)/E^ArcSinh[a*x]] + 192*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]] -
48*Pi^2*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*ArcSinh[a*x]*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*PolyL
og[3, (-I)/E^ArcSinh[a*x]] + 384*ArcSinh[a*x]*PolyLog[3, (-I)/E^ArcSinh[a*x]] - 384*ArcSinh[a*x]*PolyLog[3, (-
I)*E^ArcSinh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)/E^ArcSinh[a*x]] + 384*PolyL
og[4, (-I)*E^ArcSinh[a*x]]))/(a*c)

________________________________________________________________________________________

Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3/(a^2*c*x^2+c),x)

[Out]

int(arcsinh(a*x)^3/(a^2*c*x^2+c),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^3/(a^2*c*x^2 + c), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{asinh}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/(a**2*c*x**2+c),x)

[Out]

Integral(asinh(a*x)**3/(a**2*x**2 + 1), x)/c

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c), x)