\(\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx\) [145]

Optimal result

Integrand size = 23, antiderivative size = 261 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \]

[Out]

-b^2*(a+b*arcsinh(d*x+c))/d/e^4/(d*x+c)-1/3*(a+b*arcsinh(d*x+c))^3/d/e^4/(d*x+c)^3+b*(a+b*arcsinh(d*x+c))^2*ar
ctanh(d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4-b^3*arctanh((1+(d*x+c)^2)^(1/2))/d/e^4+b^2*(a+b*arcsinh(d*x+c))*polylog
(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4-b^2*(a+b*arcsinh(d*x+c))*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4-b^3*p
olylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4+b^3*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4-1/2*b*(a+b*arcsinh(d
*x+c))^2*(1+(d*x+c)^2)^(1/2)/d/e^4/(d*x+c)^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5859, 12, 5776, 5809, 5816, 4267, 2611, 2320, 6724, 272, 65, 213} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {b \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^4}+\frac {b^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^4}-\frac {b^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}-\frac {b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )}{d e^4} \]

[In]

Int[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^4,x]

[Out]

-((b^2*(a + b*ArcSinh[c + d*x]))/(d*e^4*(c + d*x))) - (b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(2*
d*e^4*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])^3/(3*d*e^4*(c + d*x)^3) + (b*(a + b*ArcSinh[c + d*x])^2*ArcTanh[
E^ArcSinh[c + d*x]])/(d*e^4) - (b^3*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(d*e^4) + (b^2*(a + b*ArcSinh[c + d*x])*Po
lyLog[2, -E^ArcSinh[c + d*x]])/(d*e^4) - (b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^4)
- (b^3*PolyLog[3, -E^ArcSinh[c + d*x]])/(d*e^4) + (b^3*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2320

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 2611

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 4267

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 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5809

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5816

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

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 6724

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^4}+\frac {b^2 \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^2} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\text {arcsinh}(c+d x)\right )}{2 d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{2 d e^4} \\ & = -\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+(c+d x)^2}\right )}{d e^4}-\frac {b^3 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^4} \\ & = -\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \\ & = -\frac {b^2 (a+b \text {arcsinh}(c+d x))}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^4}-\frac {b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^4}+\frac {b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^4} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(694\) vs. \(2(261)=522\).

Time = 7.25 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.66 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {a^3}{3 d e^4 (c+d x)^3}-\frac {a^2 b \sqrt {1+c^2+2 c d x+d^2 x^2}}{2 d e^4 (c+d x)^2}-\frac {a^2 b \text {arcsinh}(c+d x)}{d e^4 (c+d x)^3}-\frac {a^2 b \log (c+d x)}{2 d e^4}+\frac {a^2 b \log \left (1+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )}{2 d e^4}+\frac {a b^2 \left (-8 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-\frac {2 \left (-2+4 \text {arcsinh}(c+d x)^2+2 \cosh (2 \text {arcsinh}(c+d x))-3 (c+d x) \text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+3 (c+d x) \text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )-4 (c+d x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+2 \text {arcsinh}(c+d x) \sinh (2 \text {arcsinh}(c+d x))+\text {arcsinh}(c+d x) \log \left (1-e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))-\text {arcsinh}(c+d x) \log \left (1+e^{-\text {arcsinh}(c+d x)}\right ) \sinh (3 \text {arcsinh}(c+d x))\right )}{(c+d x)^3}\right )}{8 d e^4}+\frac {b^3 \left (-24 \text {arcsinh}(c+d x) \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )+4 \text {arcsinh}(c+d x)^3 \coth \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-6 \text {arcsinh}(c+d x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-(c+d x) \text {arcsinh}(c+d x)^3 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-24 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )+48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-\frac {16 \text {arcsinh}(c+d x)^3 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )}{(c+d x)^3}+24 \text {arcsinh}(c+d x) \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )-4 \text {arcsinh}(c+d x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )}{48 d e^4} \]

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^4,x]

[Out]

-1/3*a^3/(d*e^4*(c + d*x)^3) - (a^2*b*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d*e^4*(c + d*x)^2) - (a^2*b*ArcSin
h[c + d*x])/(d*e^4*(c + d*x)^3) - (a^2*b*Log[c + d*x])/(2*d*e^4) + (a^2*b*Log[1 + Sqrt[1 + c^2 + 2*c*d*x + d^2
*x^2]])/(2*d*e^4) + (a*b^2*(-8*PolyLog[2, -E^(-ArcSinh[c + d*x])] - (2*(-2 + 4*ArcSinh[c + d*x]^2 + 2*Cosh[2*A
rcSinh[c + d*x]] - 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])] + 3*(c + d*x)*ArcSinh[c + d*x]*
Log[1 + E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-ArcSinh[c + d*x])] + 2*ArcSinh[c + d*x]*Sinh[2*A
rcSinh[c + d*x]] + ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]] - ArcSinh[c + d*x]
*Log[1 + E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]]))/(c + d*x)^3))/(8*d*e^4) + (b^3*(-24*ArcSinh[c + d*x
]*Coth[ArcSinh[c + d*x]/2] + 4*ArcSinh[c + d*x]^3*Coth[ArcSinh[c + d*x]/2] - 6*ArcSinh[c + d*x]^2*Csch[ArcSinh
[c + d*x]/2]^2 - (c + d*x)*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d*x]/2]^4 - 24*ArcSinh[c + d*x]^2*Log[1 - E^(-A
rcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 48*Log[Tanh[ArcSinh[c + d*x]/2]] -
48*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 48*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])]
- 48*PolyLog[3, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[3, E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]^2*Sech[Ar
cSinh[c + d*x]/2]^2 - (16*ArcSinh[c + d*x]^3*Sinh[ArcSinh[c + d*x]/2]^4)/(c + d*x)^3 + 24*ArcSinh[c + d*x]*Tan
h[ArcSinh[c + d*x]/2] - 4*ArcSinh[c + d*x]^3*Tanh[ArcSinh[c + d*x]/2]))/(48*d*e^4)

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.86

method	result	size
derivativedivides	\(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\)	\(486\)
default	\(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4}}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\)	\(486\)
parts	\(-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+2 \operatorname {arcsinh}\left (d x +c \right )^{2}+6 \left (d x +c \right )^{2}\right )}{6 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}+\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2}-\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \,\operatorname {arctanh}\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{4} d}+\frac {3 a \,b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4} d}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4} d}\)	\(494\)

[In]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3*a^3/e^4/(d*x+c)^3+b^3/e^4*(-1/6/(d*x+c)^3*arcsinh(d*x+c)*(3*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+
c)+2*arcsinh(d*x+c)^2+6*(d*x+c)^2)+1/2*arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+arcsinh(d*x+c)*polylog
(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))-1/2*arcsinh(d*x+c)^2*ln(1-d*x-c-(1+(d*x+c
)^2)^(1/2))-arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))+polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))-2*arctanh
(d*x+c+(1+(d*x+c)^2)^(1/2)))+3*a*b^2/e^4*(-1/3*((1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)+arcsinh(d*x+c)^2+(d
*x+c)^2)/(d*x+c)^3+1/3*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+1/3*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2)
)-1/3*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))-1/3*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+3*a^2*b/e^4*(-1
/3/(d*x+c)^3*arcsinh(d*x+c)-1/6/(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+1/6*arctanh(1/(1+(d*x+c)^2)^(1/2))))

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="fricas")

[Out]

integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3)/(d^4*e^4*x^4 +
4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]

[In]

integrate((a+b*asinh(d*x+c))**3/(d*e*x+c*e)**4,x)

[Out]

(Integral(a**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(b**3*asinh(c
+ d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(3*a*b**2*asinh(c +
d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(3*a**2*b*asinh(c +
d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e**4

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="maxima")

[Out]

-1/3*b^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x +
c^3*d*e^4) - 1/3*a^3/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(((3*(c^3 + c)*
a*b^2 + (c^3 + c)*b^3 + (3*a*b^2*d^3 + b^3*d^3)*x^3 + 3*(3*a*b^2*c*d^2 + b^3*c*d^2)*x^2 + (3*(3*c^2*d + d)*a*b
^2 + (3*c^2*d + d)*b^3)*x + (b^3*c^2 + 3*(c^2 + 1)*a*b^2 + (3*a*b^2*d^2 + b^3*d^2)*x^2 + 2*(3*a*b^2*c*d + b^3*
c*d)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 3*(a^2*b*d^3*x
^3 + 3*a^2*b*c*d^2*x^2 + (3*c^2*d + d)*a^2*b*x + (c^3 + c)*a^2*b + (a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + (c^2 + 1)*
a^2*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))/(d^7*e^4*x^7 + 7*c
*d^6*e^4*x^6 + c^7*e^4 + c^5*e^4 + (21*c^2*d^5*e^4 + d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 + c*d^4*e^4)*x^4 + 5*(7*c
^4*d^3*e^4 + 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*x^2 + (7*c^6*d*e^4 + 5*c^4*d*e^4)*x + (d^6
*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 + c^4*e^4 + (15*c^2*d^4*e^4 + d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 + c*d^3*e^4
)*x^3 + 3*(5*c^4*d^2*e^4 + 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 + 2*c^3*d*e^4)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)), x)

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{4}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^4,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^3/(d*e*x + c*e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]

[In]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^4,x)

[Out]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^4, x)