\(\int \frac {1}{a+b \text {arccosh}(c x)} \, dx\) [536]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=-\frac {\text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c} \]

[Out]

cosh(a/b)*Shi((a+b*arccosh(c*x))/b)/b/c-Chi((a+b*arccosh(c*x))/b)*sinh(a/b)/b/c

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5881, 3384, 3379, 3382} \[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c} \]

[In]

Int[(a + b*ArcCosh[c*x])^(-1),x]

[Out]

-((CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c)) + (Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(
b*c)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 5881

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sinh[-a/b + x/b], x], x
, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c} \\ & = \frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c} \\ & = -\frac {\text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=-\frac {\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{b c} \]

[In]

Integrate[(a + b*ArcCosh[c*x])^(-1),x]

[Out]

-((CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b*c))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}}{c}\)	\(56\)
default	\(\frac {\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}}{c}\)	\(56\)

[In]

int(1/(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/2/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-1/2/b*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b))

Fricas [F]

\[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(1/(b*arccosh(c*x) + a), x)

Sympy [F]

\[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\int \frac {1}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]

[In]

integrate(1/(a+b*acosh(c*x)),x)

[Out]

Integral(1/(a + b*acosh(c*x)), x)

Maxima [F]

\[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate(1/(b*arccosh(c*x) + a), x)

Giac [F]

\[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {1}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate(1/(b*arccosh(c*x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{a+b \text {arccosh}(c x)} \, dx=\int \frac {1}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[In]

int(1/(a + b*acosh(c*x)),x)

[Out]

int(1/(a + b*acosh(c*x)), x)