\(\int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx\) [152]

Optimal result

Integrand size = 23, antiderivative size = 234 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \]

[Out]

-(a+b*arcsinh(d*x+c))^4/d/e^2/(d*x+c)-8*b*(a+b*arcsinh(d*x+c))^3*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2-12*b
^2*(a+b*arcsinh(d*x+c))^2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^2+12*b^2*(a+b*arcsinh(d*x+c))^2*polylog(2,
d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2+24*b^3*(a+b*arcsinh(d*x+c))*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^2-24*b^
3*(a+b*arcsinh(d*x+c))*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2-24*b^4*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))
/d/e^2+24*b^4*polylog(4,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5776, 5816, 4267, 2611, 6744, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {8 b \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3}{d e^2}+\frac {24 b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^2}-\frac {24 b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^2}-\frac {12 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^2}+\frac {12 b^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^2}-\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {24 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \]

[In]

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^2,x]

[Out]

-((a + b*ArcSinh[c + d*x])^4/(d*e^2*(c + d*x))) - (8*b*(a + b*ArcSinh[c + d*x])^3*ArcTanh[E^ArcSinh[c + d*x]])
/(d*e^2) - (12*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^2) + (12*b^2*(a + b*ArcSin
h[c + d*x])^2*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^2) + (24*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, -E^ArcSinh
[c + d*x]])/(d*e^2) - (24*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^2) - (24*b^4*PolyL
og[4, -E^ArcSinh[c + d*x]])/(d*e^2) + (24*b^4*PolyLog[4, E^ArcSinh[c + d*x]])/(d*e^2)

Rule 12

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

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 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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{e^2 x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{x^2} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int (a+b x)^3 \text {csch}(x) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {\left (24 b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2}-\frac {\left (24 b^3\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {\left (24 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2}+\frac {\left (24 b^4\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {\left (24 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {\left (24 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(510\) vs. \(2(234)=468\).

Time = 1.32 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\frac {-\frac {2 a^4}{c+d x}-8 a^3 b \left (\frac {\text {arcsinh}(c+d x)}{c+d x}+\log \left (\frac {1}{2} (c+d x) \text {csch}\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )+12 a^2 b^2 \left (\text {arcsinh}(c+d x) \left (-\frac {\text {arcsinh}(c+d x)}{c+d x}+2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )\right )+8 a b^3 \left (-\frac {\text {arcsinh}(c+d x)^3}{c+d x}+3 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-3 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )-6 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )\right )+b^4 \left (\pi ^4-2 \text {arcsinh}(c+d x)^4-\frac {2 \text {arcsinh}(c+d x)^4}{c+d x}-8 \text {arcsinh}(c+d x)^3 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+8 \text {arcsinh}(c+d x)^3 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )-48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )\right )}{2 d e^2} \]

[In]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^2,x]

[Out]

((-2*a^4)/(c + d*x) - 8*a^3*b*(ArcSinh[c + d*x]/(c + d*x) + Log[((c + d*x)*Csch[ArcSinh[c + d*x]/2])/2] - Log[
Sinh[ArcSinh[c + d*x]/2]]) + 12*a^2*b^2*(ArcSinh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSi
nh[c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 2*PolyLog[2, E^(-Ar
cSinh[c + d*x])]) + 8*a*b^3*(-(ArcSinh[c + d*x]^3/(c + d*x)) + 3*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*
x])] - 3*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 6*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*
x])] - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] + 6*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 6*PolyLog
[3, E^(-ArcSinh[c + d*x])]) + b^4*(Pi^4 - 2*ArcSinh[c + d*x]^4 - (2*ArcSinh[c + d*x]^4)/(c + d*x) - 8*ArcSinh[
c + d*x]^3*Log[1 + E^(-ArcSinh[c + d*x])] + 8*ArcSinh[c + d*x]^3*Log[1 - E^ArcSinh[c + d*x]] + 24*ArcSinh[c +
d*x]^2*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*PolyLog[2, E^ArcSinh[c + d*x]] + 48*ArcSinh[
c + d*x]*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 48*ArcSinh[c + d*x]*PolyLog[3, E^ArcSinh[c + d*x]] + 48*PolyLog[
4, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[4, E^ArcSinh[c + d*x]]))/(2*d*e^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(629\) vs. \(2(299)=598\).

Time = 0.47 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.69

method	result	size
derivativedivides	\(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\)	\(630\)
default	\(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\)	\(630\)
parts	\(-\frac {a^{4}}{e^{2} \left (d x +c \right ) d}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\)	\(641\)

[In]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-a^4/e^2/(d*x+c)+b^4/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^4-4*arcsinh(d*x+c)^3*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-
12*arcsinh(d*x+c)^2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+24*arcsinh(d*x+c)*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/
2))-24*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))+4*arcsinh(d*x+c)^3*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+12*arcsinh(d*x
+c)^2*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))-24*arcsinh(d*x+c)*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))+24*polylog(4
,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*a*b^3/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^3-3*arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)
^2)^(1/2))-6*arcsinh(d*x+c)*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+6*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+3*ar
csinh(d*x+c)^2*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+6*arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))-6*polylog
(3,d*x+c+(1+(d*x+c)^2)^(1/2)))+6*a^2*b^2/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^2-2*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+
c)^2)^(1/2))-2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+2*polylo
g(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*b*a^3/e^2*(-1/(d*x+c)*arcsinh(d*x+c)-arctanh(1/(1+(d*x+c)^2)^(1/2))))

Fricas [F]

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

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e)**2,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]

[In]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^2,x)

[Out]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^2, x)