\(\int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx\) [169]

Optimal result

Integrand size = 23, antiderivative size = 247 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}+\frac {e^3 \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^3 d} \]

[Out]

-3/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsinh(d*x+c))-2*e^3*(d*x+c)^4/b^2/d/(a+b*arcsinh(d*x+c))-1/2*e^3*cosh(2*a/b)*S
hi(2*(a+b*arcsinh(d*x+c))/b)/b^3/d+e^3*cosh(4*a/b)*Shi(4*(a+b*arcsinh(d*x+c))/b)/b^3/d+1/2*e^3*Chi(2*(a+b*arcs
inh(d*x+c))/b)*sinh(2*a/b)/b^3/d-e^3*Chi(4*(a+b*arcsinh(d*x+c))/b)*sinh(4*a/b)/b^3/d-1/2*e^3*(d*x+c)^3*(1+(d*x
+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^2

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5779, 5818, 5780, 5556, 3384, 3379, 3382} \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^3 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^3 d}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^3 d}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{2 b d (a+b \text {arcsinh}(c+d x))^2} \]

[In]

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

[Out]

-1/2*(e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x])^2) - (3*e^3*(c + d*x)^2)/(2*b^2*d*(
a + b*ArcSinh[c + d*x])) - (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcSinh[c + d*x])) + (e^3*CoshIntegral[(2*(a + b*
ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b^3*d) - (e^3*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(4*a)/
b])/(b^3*d) - (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b^3*d) + (e^3*Cosh[(4*a)/b]*
SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(b^3*d)

Rule 12

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

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5779

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

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5818

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

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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^3 x^3}{(a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}+\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{2 b d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}+\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x}{a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^3}{a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{b^2 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {e^3 \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b^3 d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}+\frac {\left (3 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b^3 d}-\frac {\left (2 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}+\frac {\left (e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (3 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b^3 d}+\frac {\left (2 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d}-\frac {\left (e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b^3 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}+\frac {e^3 \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.72 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\frac {e^3 \left (-\frac {b^2 (c+d x)^3 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^2}+\frac {b \left (-3 (c+d x)^2-4 (c+d x)^4\right )}{a+b \text {arcsinh}(c+d x)}+\text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )}{2 b^3 d} \]

[In]

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

[Out]

(e^3*(-((b^2*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2) + (b*(-3*(c + d*x)^2 - 4*(c + d*x)
^4))/(a + b*ArcSinh[c + d*x]) + CoshIntegral[2*(a/b + ArcSinh[c + d*x])]*Sinh[(2*a)/b] - 2*CoshIntegral[4*(a/b
+ ArcSinh[c + d*x])]*Sinh[(4*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])] + 2*Cosh[(4*a)/b]
*SinhIntegral[4*(a/b + ArcSinh[c + d*x])]))/(2*b^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(578\) vs. \(2(237)=474\).

Time = 1.02 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(\frac {-\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\operatorname {arcsinh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}+\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3} \left (2 b \,\operatorname {arcsinh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\)	\(579\)
default	\(\frac {-\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\operatorname {arcsinh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}+\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3} \left (2 b \,\operatorname {arcsinh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{32 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\)	\(579\)

[In]

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

[Out]

1/d*(-1/32*(-8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+8*(d*x+c)^4-4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+8*(d*x+c)^2+1)*e^3*(4*b
*arcsinh(d*x+c)+4*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)+1/2*e^3/b^3*exp(4*a/b)*Ei(1,4*arcsi
nh(d*x+c)+4*a/b)+1/16*(-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*(d*x+c)^2+1)*e^3*(2*b*arcsinh(d*x+c)+2*a-b)/b^2/(b^2*a
rcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)-1/4*e^3/b^3*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)+1/16/b*e^3*(2*(d
*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2+1/8/b^2*e^3*(2*(d*x+c)^2+1+2*(d*x+c)*(1+(d*x+c
)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/4/b^3*e^3*exp(-2*a/b)*Ei(1,-2*arcsinh(d*x+c)-2*a/b)-1/32/b*e^3*(8*(d*x+c)^4
+8*(d*x+c)^2+8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1)/(a+b*arcsinh(d*x+c))^2-1/8/b^2*e
^3*(8*(d*x+c)^4+8*(d*x+c)^2+8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1)/(a+b*arcsinh(d*x+
c))-1/2/b^3*e^3*exp(-4*a/b)*Ei(1,-4*arcsinh(d*x+c)-4*a/b))

Fricas [F]

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

[In]

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

[Out]

integral((d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3)/(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), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/2*((4*a*d^10*e^3 + b*d^10*e^3)*x^10 + 10*(4*a*c*d^9*e^3 + b*c*d^9*e^3)*x^9 + 3*(4*(15*c^2*d^8*e^3 + d^8*e^3
)*a + (15*c^2*d^8*e^3 + d^8*e^3)*b)*x^8 + 24*(4*(5*c^3*d^7*e^3 + c*d^7*e^3)*a + (5*c^3*d^7*e^3 + c*d^7*e^3)*b)
*x^7 + 3*(4*(70*c^4*d^6*e^3 + 28*c^2*d^6*e^3 + d^6*e^3)*a + (70*c^4*d^6*e^3 + 28*c^2*d^6*e^3 + d^6*e^3)*b)*x^6
+ 6*(4*(42*c^5*d^5*e^3 + 28*c^3*d^5*e^3 + 3*c*d^5*e^3)*a + (42*c^5*d^5*e^3 + 28*c^3*d^5*e^3 + 3*c*d^5*e^3)*b)
*x^5 + (4*(210*c^6*d^4*e^3 + 210*c^4*d^4*e^3 + 45*c^2*d^4*e^3 + d^4*e^3)*a + (210*c^6*d^4*e^3 + 210*c^4*d^4*e^
3 + 45*c^2*d^4*e^3 + d^4*e^3)*b)*x^4 + 4*(4*(30*c^7*d^3*e^3 + 42*c^5*d^3*e^3 + 15*c^3*d^3*e^3 + c*d^3*e^3)*a +
(30*c^7*d^3*e^3 + 42*c^5*d^3*e^3 + 15*c^3*d^3*e^3 + c*d^3*e^3)*b)*x^3 + 3*(4*(15*c^8*d^2*e^3 + 28*c^6*d^2*e^3
+ 15*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*a + (15*c^8*d^2*e^3 + 28*c^6*d^2*e^3 + 15*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*b)*x
^2 + ((4*a*d^7*e^3 + b*d^7*e^3)*x^7 + 7*(4*a*c*d^6*e^3 + b*c*d^6*e^3)*x^6 + (6*(14*c^2*d^5*e^3 + d^5*e^3)*a +
(21*c^2*d^5*e^3 + d^5*e^3)*b)*x^5 + 5*(2*(14*c^3*d^4*e^3 + 3*c*d^4*e^3)*a + (7*c^3*d^4*e^3 + c*d^4*e^3)*b)*x^4
+ (2*(70*c^4*d^3*e^3 + 30*c^2*d^3*e^3 + d^3*e^3)*a + 5*(7*c^4*d^3*e^3 + 2*c^2*d^3*e^3)*b)*x^3 + (6*(14*c^5*d^
2*e^3 + 10*c^3*d^2*e^3 + c*d^2*e^3)*a + (21*c^5*d^2*e^3 + 10*c^3*d^2*e^3)*b)*x^2 + 2*(2*c^7*e^3 + 3*c^5*e^3 +
c^3*e^3)*a + (c^7*e^3 + c^5*e^3)*b + (2*(14*c^6*d*e^3 + 15*c^4*d*e^3 + 3*c^2*d*e^3)*a + (7*c^6*d*e^3 + 5*c^4*d
*e^3)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (3*(4*a*d^8*e^3 + b*d^8*e^3)*x^8 + 24*(4*a*c*d^7*e^3 + b*c*d
^7*e^3)*x^7 + (24*(14*c^2*d^6*e^3 + d^6*e^3)*a + (84*c^2*d^6*e^3 + 5*d^6*e^3)*b)*x^6 + 6*(8*(14*c^3*d^5*e^3 +
3*c*d^5*e^3)*a + (28*c^3*d^5*e^3 + 5*c*d^5*e^3)*b)*x^5 + (15*(56*c^4*d^4*e^3 + 24*c^2*d^4*e^3 + d^4*e^3)*a + (
210*c^4*d^4*e^3 + 75*c^2*d^4*e^3 + 2*d^4*e^3)*b)*x^4 + 4*(3*(56*c^5*d^3*e^3 + 40*c^3*d^3*e^3 + 5*c*d^3*e^3)*a
+ (42*c^5*d^3*e^3 + 25*c^3*d^3*e^3 + 2*c*d^3*e^3)*b)*x^3 + 3*((112*c^6*d^2*e^3 + 120*c^4*d^2*e^3 + 30*c^2*d^2*
e^3 + d^2*e^3)*a + (28*c^6*d^2*e^3 + 25*c^4*d^2*e^3 + 4*c^2*d^2*e^3)*b)*x^2 + 3*(4*c^8*e^3 + 8*c^6*e^3 + 5*c^4
*e^3 + c^2*e^3)*a + (3*c^8*e^3 + 5*c^6*e^3 + 2*c^4*e^3)*b + 2*(3*(16*c^7*d*e^3 + 24*c^5*d*e^3 + 10*c^3*d*e^3 +
c*d*e^3)*a + (12*c^7*d*e^3 + 15*c^5*d*e^3 + 4*c^3*d*e^3)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 4*(c^10*e^3 +
3*c^8*e^3 + 3*c^6*e^3 + c^4*e^3)*a + (c^10*e^3 + 3*c^8*e^3 + 3*c^6*e^3 + c^4*e^3)*b + 2*(4*(5*c^9*d*e^3 + 12*c
^7*d*e^3 + 9*c^5*d*e^3 + 2*c^3*d*e^3)*a + (5*c^9*d*e^3 + 12*c^7*d*e^3 + 9*c^5*d*e^3 + 2*c^3*d*e^3)*b)*x + (4*b
*d^10*e^3*x^10 + 40*b*c*d^9*e^3*x^9 + 12*(15*c^2*d^8*e^3 + d^8*e^3)*b*x^8 + 96*(5*c^3*d^7*e^3 + c*d^7*e^3)*b*x
^7 + 12*(70*c^4*d^6*e^3 + 28*c^2*d^6*e^3 + d^6*e^3)*b*x^6 + 24*(42*c^5*d^5*e^3 + 28*c^3*d^5*e^3 + 3*c*d^5*e^3)
*b*x^5 + 4*(210*c^6*d^4*e^3 + 210*c^4*d^4*e^3 + 45*c^2*d^4*e^3 + d^4*e^3)*b*x^4 + 16*(30*c^7*d^3*e^3 + 42*c^5*
d^3*e^3 + 15*c^3*d^3*e^3 + c*d^3*e^3)*b*x^3 + 12*(15*c^8*d^2*e^3 + 28*c^6*d^2*e^3 + 15*c^4*d^2*e^3 + 2*c^2*d^2
*e^3)*b*x^2 + 8*(5*c^9*d*e^3 + 12*c^7*d*e^3 + 9*c^5*d*e^3 + 2*c^3*d*e^3)*b*x + 2*(2*b*d^7*e^3*x^7 + 14*b*c*d^6
*e^3*x^6 + 3*(14*c^2*d^5*e^3 + d^5*e^3)*b*x^5 + 5*(14*c^3*d^4*e^3 + 3*c*d^4*e^3)*b*x^4 + (70*c^4*d^3*e^3 + 30*
c^2*d^3*e^3 + d^3*e^3)*b*x^3 + 3*(14*c^5*d^2*e^3 + 10*c^3*d^2*e^3 + c*d^2*e^3)*b*x^2 + (14*c^6*d*e^3 + 15*c^4*
d*e^3 + 3*c^2*d*e^3)*b*x + (2*c^7*e^3 + 3*c^5*e^3 + c^3*e^3)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(4*b*d
^8*e^3*x^8 + 32*b*c*d^7*e^3*x^7 + 8*(14*c^2*d^6*e^3 + d^6*e^3)*b*x^6 + 16*(14*c^3*d^5*e^3 + 3*c*d^5*e^3)*b*x^5
+ 5*(56*c^4*d^4*e^3 + 24*c^2*d^4*e^3 + d^4*e^3)*b*x^4 + 4*(56*c^5*d^3*e^3 + 40*c^3*d^3*e^3 + 5*c*d^3*e^3)*b*x
^3 + (112*c^6*d^2*e^3 + 120*c^4*d^2*e^3 + 30*c^2*d^2*e^3 + d^2*e^3)*b*x^2 + 2*(16*c^7*d*e^3 + 24*c^5*d*e^3 + 1
0*c^3*d*e^3 + c*d*e^3)*b*x + (4*c^8*e^3 + 8*c^6*e^3 + 5*c^4*e^3 + c^2*e^3)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1) +
4*(c^10*e^3 + 3*c^8*e^3 + 3*c^6*e^3 + c^4*e^3)*b + (12*b*d^9*e^3*x^9 + 108*b*c*d^8*e^3*x^8 + 6*(72*c^2*d^7*e^3
+ 5*d^7*e^3)*b*x^7 + 42*(24*c^3*d^6*e^3 + 5*c*d^6*e^3)*b*x^6 + (1512*c^4*d^5*e^3 + 630*c^2*d^5*e^3 + 25*d^5*e
^3)*b*x^5 + (1512*c^5*d^4*e^3 + 1050*c^3*d^4*e^3 + 125*c*d^4*e^3)*b*x^4 + (1008*c^6*d^3*e^3 + 1050*c^4*d^3*e^3
+ 250*c^2*d^3*e^3 + 7*d^3*e^3)*b*x^3 + (432*c^7*d^2*e^3 + 630*c^5*d^2*e^3 + 250*c^3*d^2*e^3 + 21*c*d^2*e^3)*b
*x^2 + (108*c^8*d*e^3 + 210*c^6*d*e^3 + 125*c^4*d*e^3 + 21*c^2*d*e^3)*b*x + (12*c^9*e^3 + 30*c^7*e^3 + 25*c^5*
e^3 + 7*c^3*e^3)*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)) + (3*(
4*a*d^9*e^3 + b*d^9*e^3)*x^9 + 27*(4*a*c*d^8*e^3 + b*c*d^8*e^3)*x^8 + (6*(72*c^2*d^7*e^3 + 5*d^7*e^3)*a + (108
*c^2*d^7*e^3 + 7*d^7*e^3)*b)*x^7 + 7*(6*(24*c^3*d^6*e^3 + 5*c*d^6*e^3)*a + (36*c^3*d^6*e^3 + 7*c*d^6*e^3)*b)*x
^6 + ((1512*c^4*d^5*e^3 + 630*c^2*d^5*e^3 + 25*d^5*e^3)*a + (378*c^4*d^5*e^3 + 147*c^2*d^5*e^3 + 5*d^5*e^3)*b)
*x^5 + ((1512*c^5*d^4*e^3 + 1050*c^3*d^4*e^3 + 125*c*d^4*e^3)*a + (378*c^5*d^4*e^3 + 245*c^3*d^4*e^3 + 25*c*d^
4*e^3)*b)*x^4 + ((1008*c^6*d^3*e^3 + 1050*c^4*d^3*e^3 + 250*c^2*d^3*e^3 + 7*d^3*e^3)*a + (252*c^6*d^3*e^3 + 24
5*c^4*d^3*e^3 + 50*c^2*d^3*e^3 + d^3*e^3)*b)*x^3 + ((432*c^7*d^2*e^3 + 630*c^5*d^2*e^3 + 250*c^3*d^2*e^3 + 21*
c*d^2*e^3)*a + (108*c^7*d^2*e^3 + 147*c^5*d^2*e^3 + 50*c^3*d^2*e^3 + 3*c*d^2*e^3)*b)*x^2 + (12*c^9*e^3 + 30*c^
7*e^3 + 25*c^5*e^3 + 7*c^3*e^3)*a + (3*c^9*e^3 + 7*c^7*e^3 + 5*c^5*e^3 + c^3*e^3)*b + ((108*c^8*d*e^3 + 210*c^
6*d*e^3 + 125*c^4*d*e^3 + 21*c^2*d*e^3)*a + (27*c^8*d*e^3 + 49*c^6*d*e^3 + 25*c^4*d*e^3 + 3*c^2*d*e^3)*b)*x)*s
qrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*a^2*b^2*x^4 + 4
*(5*c^3*d^4 + 3*c*d^4)*a^2*b^2*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*
d^2)*a^2*b^2*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a^2*b^2 + (b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5
)*b^4*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 + 2*c^3*d^2
+ c*d^2)*b^4*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*b^4 + (b^4*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4
*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(b^4*d^5*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 + d^3)*b^4*x^2 + 2
*(2*c^3*d^2 + c*d^2)*b^4*x + (c^4*d + c^2*d)*b^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5
*x^4 + 2*(5*c^2*d^4 + d^4)*b^4*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*b^4*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*b^4*x + (
c^5*d + 2*c^3*d + c*d)*b^4)*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 + (a^2*b^2*d^4*x^3 + 3*a^2*b^2*c*d^3*x^2 + 3*a^2*b^2*c^2*d^2*x + a^2*b^2*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 +
1)^(3/2) + 3*(a^2*b^2*d^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 + c*d^2)*a
^2*b^2*x + (c^4*d + c^2*d)*a^2*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 3*(
5*c^2*d^5 + d^5)*a*b^3*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*a*b^3*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a*b^3*x^2 + 6
*(c^5*d^2 + 2*c^3*d^2 + c*d^2)*a*b^3*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a*b^3 + (a*b^3*d^4*x^3 + 3*a*b^3*c*d^
3*x^2 + 3*a*b^3*c^2*d^2*x + a*b^3*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a*b^3*d^5*x^4 + 4*a*b^3*c*d^
4*x^3 + (6*c^2*d^3 + d^3)*a*b^3*x^2 + 2*(2*c^3*d^2 + c*d^2)*a*b^3*x + (c^4*d + c^2*d)*a*b^3)*(d^2*x^2 + 2*c*d*
x + c^2 + 1) + 3*(a*b^3*d^6*x^5 + 5*a*b^3*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)*a*b^3*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*
a*b^3*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a*b^3*x + (c^5*d + 2*c^3*d + c*d)*a*b^3)*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)) + 3*(a^2*b^2*d^6*x^5 + 5*a^2*b^2*c*d^5*x^4 + 2*(5*c^2
*d^4 + d^4)*a^2*b^2*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a^2*b^2*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a^2*b^2*x + (c^5
*d + 2*c^3*d + c*d)*a^2*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate(1/2*(16*d^11*e^3*x^11 + 176*c*d^10
*e^3*x^10 + 16*c^11*e^3 + 64*c^9*e^3 + 96*c^7*e^3 + 16*(55*c^2*d^9*e^3 + 4*d^9*e^3)*x^9 + 48*(55*c^3*d^8*e^3 +
12*c*d^8*e^3)*x^8 + 64*c^5*e^3 + 96*(55*c^4*d^7*e^3 + 24*c^2*d^7*e^3 + d^7*e^3)*x^7 + 672*(11*c^5*d^6*e^3 + 8
*c^3*d^6*e^3 + c*d^6*e^3)*x^6 + 16*c^3*e^3 + 32*(231*c^6*d^5*e^3 + 252*c^4*d^5*e^3 + 63*c^2*d^5*e^3 + 2*d^5*e^
3)*x^5 + 32*(165*c^7*d^4*e^3 + 252*c^5*d^4*e^3 + 105*c^3*d^4*e^3 + 10*c*d^4*e^3)*x^4 + 16*(165*c^8*d^3*e^3 + 3
36*c^6*d^3*e^3 + 210*c^4*d^3*e^3 + 40*c^2*d^3*e^3 + d^3*e^3)*x^3 + 4*(4*d^7*e^3*x^7 + 28*c*d^6*e^3*x^6 + 4*c^7
*e^3 + 3*c^5*e^3 + 3*(28*c^2*d^5*e^3 + d^5*e^3)*x^5 + 5*(28*c^3*d^4*e^3 + 3*c*d^4*e^3)*x^4 + 10*(14*c^4*d^3*e^
3 + 3*c^2*d^3*e^3)*x^3 + 6*(14*c^5*d^2*e^3 + 5*c^3*d^2*e^3)*x^2 + (28*c^6*d*e^3 + 15*c^4*d*e^3)*x)*(d^2*x^2 +
2*c*d*x + c^2 + 1)^2 + 16*(55*c^9*d^2*e^3 + 144*c^7*d^2*e^3 + 126*c^5*d^2*e^3 + 40*c^3*d^2*e^3 + 3*c*d^2*e^3)*
x^2 + (64*d^8*e^3*x^8 + 512*c*d^7*e^3*x^7 + 64*c^8*e^3 + 100*c^6*e^3 + 42*c^4*e^3 + 4*(448*c^2*d^6*e^3 + 25*d^
6*e^3)*x^6 + 8*(448*c^3*d^5*e^3 + 75*c*d^5*e^3)*x^5 + 3*c^2*e^3 + 2*(2240*c^4*d^4*e^3 + 750*c^2*d^4*e^3 + 21*d
^4*e^3)*x^4 + 8*(448*c^5*d^3*e^3 + 250*c^3*d^3*e^3 + 21*c*d^3*e^3)*x^3 + (1792*c^6*d^2*e^3 + 1500*c^4*d^2*e^3
+ 252*c^2*d^2*e^3 + 3*d^2*e^3)*x^2 + 2*(256*c^7*d*e^3 + 300*c^5*d*e^3 + 84*c^3*d*e^3 + 3*c*d*e^3)*x)*(d^2*x^2
+ 2*c*d*x + c^2 + 1)^(3/2) + 6*(16*d^9*e^3*x^9 + 144*c*d^8*e^3*x^8 + 16*c^9*e^3 + 38*c^7*e^3 + 30*c^5*e^3 + 2*
(288*c^2*d^7*e^3 + 19*d^7*e^3)*x^7 + 14*(96*c^3*d^6*e^3 + 19*c*d^6*e^3)*x^6 + 9*c^3*e^3 + 6*(336*c^4*d^5*e^3 +
133*c^2*d^5*e^3 + 5*d^5*e^3)*x^5 + 2*(1008*c^5*d^4*e^3 + 665*c^3*d^4*e^3 + 75*c*d^4*e^3)*x^4 + c*e^3 + (1344*
c^6*d^3*e^3 + 1330*c^4*d^3*e^3 + 300*c^2*d^3*e^3 + 9*d^3*e^3)*x^3 + 3*(192*c^7*d^2*e^3 + 266*c^5*d^2*e^3 + 100
*c^3*d^2*e^3 + 9*c*d^2*e^3)*x^2 + (144*c^8*d*e^3 + 266*c^6*d*e^3 + 150*c^4*d*e^3 + 27*c^2*d*e^3 + d*e^3)*x)*(d
^2*x^2 + 2*c*d*x + c^2 + 1) + 16*(11*c^10*d*e^3 + 36*c^8*d*e^3 + 42*c^6*d*e^3 + 20*c^4*d*e^3 + 3*c^2*d*e^3)*x
+ (64*d^10*e^3*x^10 + 640*c*d^9*e^3*x^9 + 64*c^10*e^3 + 204*c^8*e^3 + 234*c^6*e^3 + 12*(240*c^2*d^8*e^3 + 17*d
^8*e^3)*x^8 + 96*(80*c^3*d^7*e^3 + 17*c*d^7*e^3)*x^7 + 115*c^4*e^3 + 6*(2240*c^4*d^6*e^3 + 952*c^2*d^6*e^3 + 3
9*d^6*e^3)*x^6 + 12*(1344*c^5*d^5*e^3 + 952*c^3*d^5*e^3 + 117*c*d^5*e^3)*x^5 + 21*c^2*e^3 + 5*(2688*c^6*d^4*e^
3 + 2856*c^4*d^4*e^3 + 702*c^2*d^4*e^3 + 23*d^4*e^3)*x^4 + 4*(1920*c^7*d^3*e^3 + 2856*c^5*d^3*e^3 + 1170*c^3*d
^3*e^3 + 115*c*d^3*e^3)*x^3 + 3*(960*c^8*d^2*e^3 + 1904*c^6*d^2*e^3 + 1170*c^4*d^2*e^3 + 230*c^2*d^2*e^3 + 7*d
^2*e^3)*x^2 + 2*(320*c^9*d*e^3 + 816*c^7*d*e^3 + 702*c^5*d*e^3 + 230*c^3*d*e^3 + 21*c*d*e^3)*x)*sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1))/(a*b^2*d^8*x^8 + 8*a*b^2*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*a*b^2*x^6 + 8*(7*c^3*d^5 + 3*c*d
^5)*a*b^2*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*a*b^2*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*a*b^2*x^3
+ 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*a*b^2*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*a*b^2*x + (c^8
+ 4*c^6 + 6*c^4 + 4*c^2 + 1)*a*b^2 + (a*b^2*d^4*x^4 + 4*a*b^2*c*d^3*x^3 + 6*a*b^2*c^2*d^2*x^2 + 4*a*b^2*c^3*d
*x + a*b^2*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + (10*c^2*d^3 + d^3)*a*
b^2*x^3 + (10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (5*c^4*d + 3*c^2*d)*a*b^2*x + (c^5 + c^3)*a*b^2)*(d^2*x^2 + 2*c*d
*x + c^2 + 1)^(3/2) + 6*(a*b^2*d^6*x^6 + 6*a*b^2*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*a*b^2*x^4 + 4*(5*c^3*d^3 + 2
*c*d^3)*a*b^2*x^3 + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*a*b^2*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*a*b^2*x + (c^6 + 2
*c^4 + c^2)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^3*d^8*x^8 + 8*b^3*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*b^3*x^
6 + 8*(7*c^3*d^5 + 3*c*d^5)*b^3*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*b^3*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3
+ 3*c*d^3)*b^3*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*b^3*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d
)*b^3*x + (c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)*b^3 + (b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c
^3*d*x + b^3*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + (10*c^2*d^3 + d^3)*b^3*
x^3 + (10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (5*c^4*d + 3*c^2*d)*b^3*x + (c^5 + c^3)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 +
1)^(3/2) + 6*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*b^3*x^4 + 4*(5*c^3*d^3 + 2*c*d^3)*b^3*x^3
+ (15*c^4*d^2 + 12*c^2*d^2 + d^2)*b^3*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*b^3*x + (c^6 + 2*c^4 + c^2)*b^3)*(d^2*
x^2 + 2*c*d*x + c^2 + 1) + 4*(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*b^3*x^5 + 5*(7*c^3*d^4 + 3*c
*d^4)*b^3*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*b^3*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (7*
c^6*d + 15*c^4*d + 9*c^2*d + d)*b^3*x + (c^7 + 3*c^5 + 3*c^3 + c)*b^3)*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)) + 4*(a*b^2*d^7*x^7 + 7*a*b^2*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*a*b^
2*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*a*b^2*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*a*b^2*x^3 + 3*(7*c^5*d^2 + 10*c^
3*d^2 + 3*c*d^2)*a*b^2*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*a*b^2*x + (c^7 + 3*c^5 + 3*c^3 + c)*a*b^2)*sqr
t(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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