Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {\text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b d} \]
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=-\frac {\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )}{b d} \]
-((CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral [a/b + ArcCosh[c + d*x]])/(b*d))
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6410, 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.
\(\displaystyle \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx\) |
\(\Big \downarrow \) 6410 |
\(\displaystyle \frac {\int \frac {1}{a+b \text {arccosh}(c+d x)}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle \frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b d}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b d}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b d}\) |
(I*(I*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b] - I*Cosh[a/b]*Sin hIntegral[(a + b*ArcCosh[c + d*x])/b]))/(b*d)
3.2.34.3.1 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[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_) + (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 = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b}}{d}\) | \(60\) |
default | \(\frac {\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b}}{d}\) | \(60\) |
\[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {1}{a+b \text {arccosh}(c+d x)} \, dx=\int \frac {1}{a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]