Integrand size = 23, antiderivative size = 410 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {2 e^4 (c+d x)^3}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e^4 (c+d x)^2 \sqrt {1+(c+d x)^2}}{b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {25 e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{6 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e^4 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{48 b^4 d}+\frac {27 e^4 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{32 b^4 d}-\frac {125 e^4 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{96 b^4 d}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{96 b^4 d} \]
-2/3*e^4*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))^2-5/6*e^4*(d*x+c)^5/b^2/d/(a +b*arcsinh(d*x+c))^2+1/48*e^4*cosh(a/b)*Shi((a+b*arcsinh(d*x+c))/b)/b^4/d- 27/32*e^4*cosh(3*a/b)*Shi(3*(a+b*arcsinh(d*x+c))/b)/b^4/d+125/96*e^4*cosh( 5*a/b)*Shi(5*(a+b*arcsinh(d*x+c))/b)/b^4/d-1/48*e^4*Chi((a+b*arcsinh(d*x+c ))/b)*sinh(a/b)/b^4/d+27/32*e^4*Chi(3*(a+b*arcsinh(d*x+c))/b)*sinh(3*a/b)/ b^4/d-125/96*e^4*Chi(5*(a+b*arcsinh(d*x+c))/b)*sinh(5*a/b)/b^4/d-1/3*e^4*( d*x+c)^4*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^3-2*e^4*(d*x+c)^2*(1 +(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))-25/6*e^4*(d*x+c)^4*(1+(d*x+c) ^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))
Time = 1.83 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00 \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e^4 \left (\frac {32 b^3 (c+d x)^4 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^3}-\frac {16 b^2 \left (-4 (c+d x)^3-5 (c+d x)^5\right )}{(a+b \text {arcsinh}(c+d x))^2}+\frac {16 b \sqrt {1+(c+d x)^2} \left (12 (c+d x)^2+25 (c+d x)^4\right )}{a+b \text {arcsinh}(c+d x)}+384 \left (\text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+544 \left (-3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+\text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )+125 \left (10 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-5 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-10 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{96 b^4 d} \]
-1/96*(e^4*((32*b^3*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 - (16*b^2*(-4*(c + d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcSinh[c + d*x ])^2 + (16*b*Sqrt[1 + (c + d*x)^2]*(12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*ArcSinh[c + d*x]) + 384*(CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]]) + 544*(-3*CoshIntegral[ a/b + ArcSinh[c + d*x]]*Sinh[a/b] + CoshIntegral[3*(a/b + ArcSinh[c + d*x] )]*Sinh[(3*a)/b] + 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Cosh [(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]) + 125*(10*CoshIntegral [a/b + ArcSinh[c + d*x]]*Sinh[a/b] - 5*CoshIntegral[3*(a/b + ArcSinh[c + d *x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcSinh[c + d*x])]*Sinh[(5*a)/ b] - 10*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 5*Cosh[(3*a)/b]*S inhIntegral[3*(a/b + ArcSinh[c + d*x])] - Cosh[(5*a)/b]*SinhIntegral[5*(a/ b + ArcSinh[c + d*x])])))/(b^4*d)
Time = 1.12 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6274, 27, 6194, 6233, 6193, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{3 b}+\frac {5 \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \left (\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {5 \left (\frac {5 \int \frac {(c+d x)^4}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^4 \left (\frac {5 \left (\frac {5 \left (\frac {\int \left (-\frac {5 \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 (a+b \text {arcsinh}(c+d x))}+\frac {9 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{16 (a+b \text {arcsinh}(c+d x))}-\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^4 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {4 \left (\frac {3 \left (\frac {\int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {4 \left (\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {5 \left (\frac {5 \left (\frac {-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {9}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {5}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {9}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {5}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^4 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^5}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^4}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
(e^4*(-1/3*((c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]) ^3) + (4*(-1/2*(c + d*x)^3/(b*(a + b*ArcSinh[c + d*x])^2) + (3*(-(((c + d* x)^2*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]))) + ((CoshIntegral [(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/4 - (3*CoshIntegral[(3*(a + b*ArcS inh[c + d*x]))/b]*Sinh[(3*a)/b])/4 - (Cosh[a/b]*SinhIntegral[(a + b*ArcSin h[c + d*x])/b])/4 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d* x]))/b])/4)/b^2))/(2*b)))/(3*b) + (5*(-1/2*(c + d*x)^5/(b*(a + b*ArcSinh[c + d*x])^2) + (5*(-(((c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[ c + d*x]))) + (-1/8*(CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b]) + (9*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/16 - (5*Co shIntegral[(5*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]* SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/8 - (9*Cosh[(3*a)/b]*SinhIntegra l[(3*(a + b*ArcSinh[c + d*x]))/b])/16 + (5*Cosh[(5*a)/b]*SinhIntegral[(5*( a + b*ArcSinh[c + d*x]))/b])/16)/b^2))/(2*b)))/(3*b)))/d
3.2.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[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] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
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] - Simp[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]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1243\) vs. \(2(384)=768\).
Time = 0.90 (sec) , antiderivative size = 1244, normalized size of antiderivative = 3.03
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1244\) |
| default | \(\text {Expression too large to display}\) | \(1244\) |
1/d*(1/192*(-16*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+16*(d*x+c)^5-12*(d*x+c)^2*(1 +(d*x+c)^2)^(1/2)+20*(d*x+c)^3-(1+(d*x+c)^2)^(1/2)+5*d*x+5*c)*e^4*(25*b^2* arcsinh(d*x+c)^2+50*a*b*arcsinh(d*x+c)-5*b^2*arcsinh(d*x+c)+25*a^2-5*a*b+2 *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)+125/192*e^4/b^4*exp(5*a/b)*Ei(1,5*arcsinh(d*x+c)+5*a/b)-1/64*(- 4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x+c)^2)^(1/2)+3*d*x+3*c) *e^4*(9*b^2*arcsinh(d*x+c)^2+18*a*b*arcsinh(d*x+c)-3*b^2*arcsinh(d*x+c)+9* a^2-3*a*b+2*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)-27/64*e^4/b^4*exp(3*a/b)*Ei(1,3*arcsinh(d*x+c)+3*a/b )+1/96*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^4*(b^2*arcsinh(d*x+c)^2+2*a*b*arcsin h(d*x+c)-b^2*arcsinh(d*x+c)+a^2-a*b+2*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)+1/96*e^4/b^4*exp(a/b)*Ei(1 ,arcsinh(d*x+c)+a/b)-1/48/b*e^4*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d *x+c))^3-1/96/b^2*e^4*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-1 /96/b^3*e^4*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-1/96/b^4*e^4* exp(-a/b)*Ei(1,-arcsinh(d*x+c)-a/b)+1/32/b*e^4*(4*(d*x+c)^3+3*d*x+3*c+4*(d *x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^3+3/ 64/b^2*e^4*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+ c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2+9/64/b^3*e^4*(4*(d*x+c)^3+3*d*x+3*c+4* (d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))...
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
integral((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)/(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)
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(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) + Int egral(d**4*x**4/(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) + Integra l(4*c*d**3*x**3/(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) + Integra l(6*c**2*d**2*x**2/(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) + Inte gral(4*c**3*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))
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]