Integrand size = 10, antiderivative size = 59 \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{4 a^3}+\frac {3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3} \] Output:
-x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)+1/4*Chi(arccosh(a*x))/a^3+ 3/4*Chi(3*arccosh(a*x))/a^3
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\frac {-\frac {4 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\text {arccosh}(a x)}+\text {Chi}(\text {arccosh}(a x))+3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3} \] Input:
Integrate[x^2/ArcCosh[a*x]^2,x]
Output:
((-4*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))/ArcCosh[a*x] + CoshInte gral[ArcCosh[a*x]] + 3*CoshIntegral[3*ArcCosh[a*x]])/(4*a^3)
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6300, 2009}
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.
\(\displaystyle \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle -\frac {\int \left (-\frac {a x}{4 \text {arccosh}(a x)}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\) |
Input:
Int[x^2/ArcCosh[a*x]^2,x]
Output:
-((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) - (-1/4*CoshIntegra l[ArcCosh[a*x]] - (3*CoshIntegral[3*ArcCosh[a*x]])/4)/a^3
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 = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{4}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4}}{a^{3}}\) | \(59\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{4}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4}}{a^{3}}\) | \(59\) |
Input:
int(x^2/arccosh(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a^3*(-1/4/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/4*Chi(arccosh(a*x)) -1/4/arccosh(a*x)*sinh(3*arccosh(a*x))+3/4*Chi(3*arccosh(a*x)))
\[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^2,x, algorithm="fricas")
Output:
integral(x^2/arccosh(a*x)^2, x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/acosh(a*x)**2,x)
Output:
Integral(x**2/acosh(a*x)**2, x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^2,x, algorithm="maxima")
Output:
-(a^3*x^5 - a*x^3 + (a^2*x^4 - x^2)*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((3*a^5*x^6 - 6*a^3*x^4 + (3*a^3*x^4 - a*x^2)*(a*x + 1) *(a*x - 1) + 3*a*x^2 + (6*a^4*x^5 - 7*a^2*x^3 + 2*x)*sqrt(a*x + 1)*sqrt(a* x - 1))/((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)
\[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^2,x, algorithm="giac")
Output:
integrate(x^2/arccosh(a*x)^2, x)
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \] Input:
int(x^2/acosh(a*x)^2,x)
Output:
int(x^2/acosh(a*x)^2, x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^{2}}{\mathit {acosh} \left (a x \right )^{2}}d x \] Input:
int(x^2/acosh(a*x)^2,x)
Output:
int(x**2/acosh(a*x)**2,x)