Integrand size = 6, antiderivative size = 39 \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{a} \] Output:
-(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)+Chi(arccosh(a*x))/a
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\frac {1-a x+\sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))}{a \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)} \] Input:
Integrate[ArcCosh[a*x]^(-2),x]
Output:
(1 - a*x + Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*CoshIntegral[ArcCosh[a* x]])/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x])
Time = 0.61 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6295, 6368, 3042, 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.
\(\displaystyle \int \frac {1}{\text {arccosh}(a x)^2} \, dx\) |
\(\Big \downarrow \) 6295 |
\(\displaystyle a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {\int \frac {a x}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\text {Chi}(\text {arccosh}(a x))}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
Input:
Int[ArcCosh[a*x]^(-2),x]
Output:
-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[ a*x]]/a
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[((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]
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{a}\) | \(33\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{a}\) | \(33\) |
Input:
int(1/arccosh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(-1/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+Chi(arccosh(a*x)))
\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(1/arccosh(a*x)^2,x, algorithm="fricas")
Output:
integral(arccosh(a*x)^(-2), x)
\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \] Input:
integrate(1/acosh(a*x)**2,x)
Output:
Integral(acosh(a*x)**(-2), x)
\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(1/arccosh(a*x)^2,x, algorithm="maxima")
Output:
-(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x)/((a^3*x^2 + s qrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1 ))) + integrate((a^4*x^4 - 2*a^2*x^2 + (a^2*x^2 + 1)*(a*x + 1)*(a*x - 1) + (2*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)/((a^4*x^4 + (a*x + 1)* (a*x - 1)*a^2*x^2 - 2*a^2*x^2 + 2*(a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)
\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(1/arccosh(a*x)^2,x, algorithm="giac")
Output:
integrate(arccosh(a*x)^(-2), x)
Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \] Input:
int(1/acosh(a*x)^2,x)
Output:
int(1/acosh(a*x)^2, x)
\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int \frac {1}{\mathit {acosh} \left (a x \right )^{2}}d x \] Input:
int(1/acosh(a*x)^2,x)
Output:
int(1/acosh(a*x)**2,x)