Integrand size = 8, antiderivative size = 42 \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2} \] Output:
-x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)+Chi(2*arccosh(a*x))/a^2
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\frac {-\frac {a x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\text {arccosh}(a x)}+\text {Chi}(2 \text {arccosh}(a x))}{a^2} \] Input:
Integrate[x/ArcCosh[a*x]^2,x]
Output:
(-((a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))/ArcCosh[a*x]) + CoshIntegral [2*ArcCosh[a*x]])/a^2
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6300, 25, 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.
\(\displaystyle \int \frac {x}{\text {arccosh}(a x)^2} \, dx\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle -\frac {\int -\frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
Input:
Int[x/ArcCosh[a*x]^2,x]
Output:
-((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[2*ArcC osh[a*x]]/a^2
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_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\) | \(28\) |
default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\) | \(28\) |
Input:
int(x/arccosh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*(-1/2*sinh(2*arccosh(a*x))/arccosh(a*x)+Chi(2*arccosh(a*x)))
\[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x/arccosh(a*x)^2,x, algorithm="fricas")
Output:
integral(x/arccosh(a*x)^2, x)
\[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int \frac {x}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \] Input:
integrate(x/acosh(a*x)**2,x)
Output:
Integral(x/acosh(a*x)**2, x)
\[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x/arccosh(a*x)^2,x, algorithm="maxima")
Output:
-(a^3*x^4 - a*x^2 + (a^2*x^3 - x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate((2*a^5*x^5 + 2*(a*x + 1)*(a*x - 1)*a^3*x^3 - 4*a^3*x^3 + (4*a^4*x^4 - 4*a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(a*x - 1) + 2*a*x)/((a^5*x^ 4 + (a*x + 1)*(a*x - 1)*a^3*x^2 - 2*a^3*x^2 + 2*(a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)
\[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x/arccosh(a*x)^2,x, algorithm="giac")
Output:
integrate(x/arccosh(a*x)^2, x)
Timed out. \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \] Input:
int(x/acosh(a*x)^2,x)
Output:
int(x/acosh(a*x)^2, x)
\[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\int \frac {x}{\mathit {acosh} \left (a x \right )^{2}}d x \] Input:
int(x/acosh(a*x)^2,x)
Output:
int(x/acosh(a*x)**2,x)