Integrand size = 21, antiderivative size = 218 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e}{6 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {e (c+d x)^2}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d} \]
1/6*e/b^2/d/(a+b*arccosh(d*x+c))^2-1/3*e*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+ c))^2+2/3*e*Chi(2*(a+b*arccosh(d*x+c))/b)*cosh(2*a/b)/b^4/d-2/3*e*Shi(2*(a +b*arccosh(d*x+c))/b)*sinh(2*a/b)/b^4/d-1/3*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x +c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^3-2/3*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+ c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c))
Time = 0.90 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e \left (-\frac {2 b^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {b^2 \left (1-2 (c+d x)^2\right )}{(a+b \text {arccosh}(c+d x))^2}-\frac {4 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{a+b \text {arccosh}(c+d x)}-4 \log (a+b \text {arccosh}(c+d x))+4 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\log (a+b \text {arccosh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \]
(e*((-2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh [c + d*x])^3 + (b^2*(1 - 2*(c + d*x)^2))/(a + b*ArcCosh[c + d*x])^2 - (4*b *Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x]) - 4*Log[a + b*ArcCosh[c + d*x]] + 4*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + A rcCosh[c + d*x])] + Log[a + b*ArcCosh[c + d*x]] - Sinh[(2*a)/b]*SinhIntegr al[2*(a/b + ArcCosh[c + d*x])])))/(6*b^4*d)
Time = 1.44 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6411, 27, 6301, 6308, 6366, 6300, 25, 3042, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {e \left (-\frac {\int \frac {1}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}+\frac {2 \int \frac {(c+d x)^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {e \left (\frac {2 \int \frac {(c+d x)^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {-\frac {\int -\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (\frac {2 \left (-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e \left (\frac {2 \left (-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+i \sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {-\frac {\sinh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (\frac {2 \left (-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e \left (\frac {2 \left (-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e \left (\frac {2 \left (-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}+\frac {1}{6 b^2 (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
(e*(-1/3*(Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*(a + b*ArcCos h[c + d*x])^3) + 1/(6*b^2*(a + b*ArcCosh[c + d*x])^2) + (2*(-1/2*(c + d*x) ^2/(b*(a + b*ArcCosh[c + d*x])^2) + (-((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[ 1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) - (-(Cosh[(2*a)/b]*CoshIntegra l[(2*(a + b*ArcCosh[c + d*x]))/b]) + Sinh[(2*a)/b]*SinhIntegral[(2*(a + b* ArcCosh[c + d*x]))/b])/b^2)/b))/(3*b)))/d
3.2.51.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(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* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.08 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.62
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\) | \(353\) |
| default | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\) | \(353\) |
1/d*(1/12*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e*(2* b^2*arccosh(d*x+c)^2+4*a*b*arccosh(d*x+c)-b^2*arccosh(d*x+c)+2*a^2-a*b+b^2 )/b^3/(b^3*arccosh(d*x+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x+c )+a^3)-1/3*e/b^4*exp(2*a/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/12/b*e*(2*(d*x+ c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^3-1 /12/b^2*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*a rccosh(d*x+c))^2-1/6/b^3*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2 )*(d*x+c))/(a+b*arccosh(d*x+c))-1/3/b^4*e*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+ c)-2*a/b))
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
integral((d*e*x + c*e)/(b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^ 3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh(d*x + c) + a^4), x)
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=e \left (\int \frac {c}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e*(Integral(c/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x) **2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Integral( d*x/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*a* b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x))
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \]