Integrand size = 12, antiderivative size = 132 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}-\frac {c+d x}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {\text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{2 b^3 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{2 b^3 d} \]
1/2*(-d*x-c)/b^2/d/(a+b*arccosh(d*x+c))+1/2*cosh(a/b)*Shi((a+b*arccosh(d*x +c))/b)/b^3/d-1/2*Chi((a+b*arccosh(d*x+c))/b)*sinh(a/b)/b^3/d-1/2*(d*x+c-1 )^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^2
Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {\frac {b \left (a c+a d x+b \sqrt {-1+c+d x} \sqrt {1+c+d x}+b (c+d x) \text {arccosh}(c+d x)\right )}{(a+b \text {arccosh}(c+d x))^2}+\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )}{2 b^3 d} \]
-1/2*((b*(a*c + a*d*x + b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*(c + d* x)*ArcCosh[c + d*x]))/(a + b*ArcCosh[c + d*x])^2 + CoshIntegral[a/b + ArcC osh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]])/ (b^3*d)
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6410, 6295, 6366, 6296, 25, 3042, 26, 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.
| \(\displaystyle \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 6410 |
| \(\displaystyle \frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}}{d}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}}{d}\) |
| \(\Big \downarrow \) 6296 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}-\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{2 b}}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{2 b}}{d}\) |
(-1/2*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x])^2 ) + (-((c + d*x)/(b*(a + b*ArcCosh[c + d*x]))) + (I*(I*CoshIntegral[(a + b *ArcCosh[c + d*x])/b]*Sinh[a/b] - I*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[ c + d*x])/b]))/b^2)/(2*b))/d
3.2.46.3.1 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[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_Symbol] :> Simp[Sqrt[1 + c* x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c /(b*(n + 1)) Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.57
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) \left (b \,\operatorname {arccosh}\left (d x +c \right )+a -b \right )}{4 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{4 b^{3}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{4 b^{3}}}{d}\) | \(207\) |
| default | \(\frac {-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) \left (b \,\operatorname {arccosh}\left (d x +c \right )+a -b \right )}{4 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{4 b^{3}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{4 b^{3}}}{d}\) | \(207\) |
1/d*(-1/4*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*(b*arccosh(d*x+c)+a-b)/ b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+1/4/b^3*exp(a/b)*Ei(1, arccosh(d*x+c)+a/b)-1/4/b*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arc cosh(d*x+c))^2-1/4/b^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccos h(d*x+c))-1/4/b^3*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b))
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integral(1/(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), x)
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}\, dx \]
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
-1/2*((a*d^7 + b*d^7)*x^7 + 7*(a*c*d^6 + b*c*d^6)*x^6 + 3*((7*c^2*d^5 - d^ 5)*a + (7*c^2*d^5 - d^5)*b)*x^5 + 5*((7*c^3*d^4 - 3*c*d^4)*a + (7*c^3*d^4 - 3*c*d^4)*b)*x^4 + ((a*d^4 + b*d^4)*x^4 + 4*(a*c*d^3 + b*c*d^3)*x^3 + (6* a*c^2*d^2 + (6*c^2*d^2 - d^2)*b)*x^2 + (c^4 - 1)*a + (c^4 - c^2)*b + 2*(2* a*c^3*d + (2*c^3*d - c*d)*b)*x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + ((35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*a + (35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*b )*x^3 + (3*(a*d^5 + b*d^5)*x^5 + 15*(a*c*d^4 + b*c*d^4)*x^4 + (3*(10*c^2*d ^3 - d^3)*a + 5*(6*c^2*d^3 - d^3)*b)*x^3 + 3*((10*c^3*d^2 - 3*c*d^2)*a + 5 *(2*c^3*d^2 - c*d^2)*b)*x^2 + 3*(c^5 - c^3)*a + (3*c^5 - 5*c^3 + 2*c)*b + (3*(5*c^4*d - 3*c^2*d)*a + (15*c^4*d - 15*c^2*d + 2*d)*b)*x)*(d*x + c + 1) *(d*x + c - 1) + 3*((7*c^5*d^2 - 10*c^3*d^2 + 3*c*d^2)*a + (7*c^5*d^2 - 10 *c^3*d^2 + 3*c*d^2)*b)*x^2 + (3*(a*d^6 + b*d^6)*x^6 + 18*(a*c*d^5 + b*c*d^ 5)*x^5 + (3*(15*c^2*d^4 - 2*d^4)*a + (45*c^2*d^4 - 7*d^4)*b)*x^4 + 4*(3*(5 *c^3*d^3 - 2*c*d^3)*a + (15*c^3*d^3 - 7*c*d^3)*b)*x^3 + ((45*c^4*d^2 - 36* c^2*d^2 + 4*d^2)*a + (45*c^4*d^2 - 42*c^2*d^2 + 5*d^2)*b)*x^2 + (3*c^6 - 6 *c^4 + 4*c^2 - 1)*a + (3*c^6 - 7*c^4 + 5*c^2 - 1)*b + 2*((9*c^5*d - 12*c^3 *d + 4*c*d)*a + (9*c^5*d - 14*c^3*d + 5*c*d)*b)*x)*sqrt(d*x + c + 1)*sqrt( d*x + c - 1) + (c^7 - 3*c^5 + 3*c^3 - c)*a + (c^7 - 3*c^5 + 3*c^3 - c)*b + ((7*c^6*d - 15*c^4*d + 9*c^2*d - d)*a + (7*c^6*d - 15*c^4*d + 9*c^2*d - d )*b)*x + (b*d^7*x^7 + 7*b*c*d^6*x^6 + 3*(7*c^2*d^5 - d^5)*b*x^5 + 5*(7*...
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \]