\(\int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^4} \, dx\) [177]

Optimal result

Integrand size = 21, antiderivative size = 204 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d} \]

[Out]

-1/6*e/b^2/d/(a+b*arcsinh(d*x+c))^2-1/3*e*(d*x+c)^2/b^2/d/(a+b*arcsinh(d*x+c))^2+2/3*e*Chi(2*(a+b*arcsinh(d*x+
c))/b)*cosh(2*a/b)/b^4/d-2/3*e*Shi(2*(a+b*arcsinh(d*x+c))/b)*sinh(2*a/b)/b^4/d-1/3*e*(d*x+c)*(1+(d*x+c)^2)^(1/
2)/b/d/(a+b*arcsinh(d*x+c))^3-2/3*e*(d*x+c)*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5859, 12, 5779, 5818, 5778, 3384, 3379, 3382, 5783} \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sqrt {(c+d x)^2+1} (c+d x)}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e \sqrt {(c+d x)^2+1} (c+d x)}{3 b d (a+b \text {arcsinh}(c+d x))^3} \]

[In]

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

[Out]

-1/3*(e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x])^3) - e/(6*b^2*d*(a + b*ArcSinh[c + d*x]
)^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (2*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*b^3*d*(
a + b*ArcSinh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d) - (2*e*S
inh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*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 5778

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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 5783

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

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 x}{(a+b \text {arcsinh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{(a+b \text {arcsinh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{3 b d}+\frac {(2 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d} \\ & = -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {\left (2 e \cosh \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 )}{3 b^4 d}-\frac {\left (2 e \sinh \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 )}{3 b^4 d} \\ & = -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e (c+d x) \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.89 \[ \int \frac {c e+d e x}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {e \left (-\frac {2 b^3 (c+d x) \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^3}+\frac {b^2 \left (-1-2 (c+d x)^2\right )}{(a+b \text {arcsinh}(c+d x))^2}-\frac {4 b (c+d x) \sqrt {1+(c+d x)^2}}{a+b \text {arcsinh}(c+d x)}+4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+4 \log (a+b \text {arcsinh}(c+d x))-4 \left (\log (a+b \text {arcsinh}(c+d x))+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \]

[In]

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

[Out]

(e*((-2*b^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 + (b^2*(-1 - 2*(c + d*x)^2))/(a + b*Ar
cSinh[c + d*x])^2 - (4*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + 4*Cosh[(2*a)/b]*CoshInteg
ral[2*(a/b + ArcSinh[c + d*x])] + 4*Log[a + b*ArcSinh[c + d*x]] - 4*(Log[a + b*ArcSinh[c + d*x]] + Sinh[(2*a)/
b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])])))/(6*b^4*d)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.63

method	result	size
derivativedivides	\(\frac {\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e \left (2 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{12 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{12 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{6 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\)	\(333\)
default	\(\frac {\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e \left (2 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{12 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{12 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{6 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\)	\(333\)

[In]

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

[Out]

1/d*(1/12*(-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*(d*x+c)^2+1)*e*(2*b^2*arcsinh(d*x+c)^2+4*a*b*arcsinh(d*x+c)-b^2*ar
csinh(d*x+c)+2*a^2-a*b+b^2)/b^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)-1/3
*e/b^4*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)-1/12/b*e*(2*(d*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arc
sinh(d*x+c))^3-1/12/b^2*e*(2*(d*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-1/6/b^3*e*(2*(d
*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-1/3/b^4*e*exp(-2*a/b)*Ei(1,-2*arcsinh(d*x+c)-2*a
/b))

Fricas [F]

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

[In]

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

[Out]

integral((d*e*x + c*e)/(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), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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