Integrand size = 12, antiderivative size = 135 \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{4 a^3} \] Output:
-2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)+1/4*Pi^(1/2)*erf(a rccosh(a*x)^(1/2))/a^3+1/4*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arccosh(a*x)^(1/2) )/a^3+1/4*Pi^(1/2)*erfi(arccosh(a*x)^(1/2))/a^3+1/4*3^(1/2)*Pi^(1/2)*erfi( 3^(1/2)*arccosh(a*x)^(1/2))/a^3
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.03 \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-\sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+\sqrt {3} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+2 \sinh (3 \text {arccosh}(a x))}{4 a^3 \sqrt {\text {arccosh}(a x)}} \] Input:
Integrate[x^2/ArcCosh[a*x]^(3/2),x]
Output:
-1/4*(2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) - Sqrt[3]*Sqrt[-ArcCosh[a*x]] *Gamma[1/2, -3*ArcCosh[a*x]] - Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x ]] + Sqrt[ArcCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]] + Sqrt[3]*Sqrt[ArcCosh[a* x]]*Gamma[1/2, 3*ArcCosh[a*x]] + 2*Sinh[3*ArcCosh[a*x]])/(a^3*Sqrt[ArcCosh [a*x]])
Time = 0.42 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
\(\displaystyle \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle -\frac {2 \int \left (-\frac {a x}{4 \sqrt {\text {arccosh}(a x)}}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\) |
Input:
Int[x^2/ArcCosh[a*x]^(3/2),x]
Output:
(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) - (2*(-1/8*(S qrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a* x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/8 - (Sqrt[3*Pi]*Erfi[Sqrt[3] *Sqrt[ArcCosh[a*x]]])/8))/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]
\[\int \frac {x^{2}}{\operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]
Input:
int(x^2/arccosh(a*x)^(3/2),x)
Output:
int(x^2/arccosh(a*x)^(3/2),x)
Exception generated. \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/arccosh(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x^{2}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(x**2/acosh(a*x)**(3/2),x)
Output:
Integral(x**2/acosh(a*x)**(3/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^2/arccosh(a*x)^(3/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^2/arccosh(a*x)^(3/2), x)
Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \] Input:
int(x^2/acosh(a*x)^(3/2),x)
Output:
int(x^2/acosh(a*x)^(3/2), x)
\[ \int \frac {x^2}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {6 \mathit {acosh} \left (a x \right ) \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{\mathit {acosh} \left (a x \right ) a^{2} x^{2}-\mathit {acosh} \left (a x \right )}d x \right ) a^{2}-4 \mathit {acosh} \left (a x \right ) \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right ) a^{2} x^{2}-\mathit {acosh} \left (a x \right )}d x \right )-2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right ) a} \] Input:
int(x^2/acosh(a*x)^(3/2),x)
Output:
(2*(3*acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**3)/( acosh(a*x)*a**2*x**2 - acosh(a*x)),x)*a**2 - 2*acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(acosh(a*x)*a**2*x**2 - acosh(a*x)),x ) - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**2))/(acosh(a*x)*a)