Integrand size = 12, antiderivative size = 98 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{b d (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2 d} \]
Chi((a+b*arccosh(d*x+c))/b)*cosh(a/b)/b^2/d-Shi((a+b*arccosh(d*x+c))/b)*si nh(a/b)/b^2/d-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))
Time = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\frac {-\frac {b \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \text {arccosh}(c+d x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )}{b^2 d} \]
(-((b*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x))/(a + b*ArcCosh[c + d*x])) + Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c + d*x]] - Sinh[a/b]*SinhI ntegral[a/b + ArcCosh[c + d*x]])/(b^2*d)
Time = 0.72 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6410, 6295, 6368, 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.
| \(\displaystyle \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 6410 |
| \(\displaystyle \frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{d}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle \frac {\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{d}\) |
(-((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b] - Sinh[a/b]*SinhIntegr al[(a + b*ArcCosh[c + d*x])/b])/b^2)/d
3.2.40.3.1 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_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
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 = 0.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\) | \(139\) |
| default | \(\frac {\frac {-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d}\) | \(139\) |
1/d*(1/2*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)/b/(a+b*arccosh(d*x+c))-1 /2/b^2*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/2/b*(d*x+c+(d*x+c-1)^(1/2)*(d*x +c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/2/b^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a /b))
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
-(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c)/(a*b*d^3*x^2 + 2*a*b*c*d^2* x + (c^2*d - d)*a*b + (a*b*d^2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d - d)*b^2 + (b^2*d^2*x + b^2* c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt (d*x + c - 1) + c)) + integrate((d^4*x^4 + 4*c*d^3*x^3 + c^4 + (d^2*x^2 + 2*c*d*x + c^2 + 1)*(d*x + c + 1)*(d*x + c - 1) + 2*(3*c^2*d^2 - d^2)*x^2 + (2*d^3*x^3 + 6*c*d^2*x^2 + 2*c^3 + (6*c^2*d - d)*x - c)*sqrt(d*x + c + 1) *sqrt(d*x + c - 1) - 2*c^2 + 4*(c^3*d - c*d)*x + 1)/(a*b*d^4*x^4 + 4*a*b*c *d^3*x^3 + 2*(3*c^2*d^2 - d^2)*a*b*x^2 + 4*(c^3*d - c*d)*a*b*x + (a*b*d^2* x^2 + 2*a*b*c*d*x + a*b*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*a*b + 2*(a*b*d^3*x^3 + 3*a*b*c*d^2*x^2 + (3*c^2*d - d)*a*b*x + (c^3 - c )*a*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^ 3 + 2*(3*c^2*d^2 - d^2)*b^2*x^2 + 4*(c^3*d - c*d)*b^2*x + (b^2*d^2*x^2 + 2 *b^2*c*d*x + b^2*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*b^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d - d)*b^2*x + (c^3 - c)*b^2)* sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)), x)
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \]