Integrand size = 12, antiderivative size = 105 \[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^3} \] Output:
-1/8*Pi^(1/2)*erf(arccosh(a*x)^(1/2))/a^3-1/24*3^(1/2)*Pi^(1/2)*erf(3^(1/2 )*arccosh(a*x)^(1/2))/a^3+1/8*Pi^(1/2)*erfi(arccosh(a*x)^(1/2))/a^3+1/24*3 ^(1/2)*Pi^(1/2)*erfi(3^(1/2)*arccosh(a*x)^(1/2))/a^3
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )+3 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (3 \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+\sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )\right )}{24 a^3 \sqrt {\text {arccosh}(a x)}} \] Input:
Integrate[x^2/Sqrt[ArcCosh[a*x]],x]
Output:
(Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -3*ArcCosh[a*x]] + 3*Sqrt[-ArcCosh [a*x]]*Gamma[1/2, -ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(3*Gamma[1/2, ArcCos h[a*x]] + Sqrt[3]*Gamma[1/2, 3*ArcCosh[a*x]]))/(24*a^3*Sqrt[ArcCosh[a*x]])
Time = 0.35 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 5971, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx\) |
\(\Big \downarrow \) 6302 |
\(\displaystyle \frac {\int \frac {a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^3}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\int \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{a^3}\) |
Input:
Int[x^2/Sqrt[ArcCosh[a*x]],x]
Output:
(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[Ar cCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi/3]*Erfi [Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8)/a^3
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
\[\int \frac {x^{2}}{\sqrt {\operatorname {arccosh}\left (a x \right )}}d x\]
Input:
int(x^2/arccosh(a*x)^(1/2),x)
Output:
int(x^2/arccosh(a*x)^(1/2),x)
Exception generated. \[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2/arccosh(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {x^{2}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \] Input:
integrate(x**2/acosh(a*x)**(1/2),x)
Output:
Integral(x**2/sqrt(acosh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {x^{2}}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/sqrt(arccosh(a*x)), x)
\[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\int { \frac {x^{2}}{\sqrt {\operatorname {arcosh}\left (a x\right )}} \,d x } \] Input:
integrate(x^2/arccosh(a*x)^(1/2),x, algorithm="giac")
Output:
integrate(x^2/sqrt(arccosh(a*x)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\int \frac {x^2}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \] Input:
int(x^2/acosh(a*x)^(1/2),x)
Output:
int(x^2/acosh(a*x)^(1/2), x)
\[ \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}-6 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{a^{2} x^{2}-1}d x \right ) a^{2}+4 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{a^{2} x^{2}-1}d x \right )}{a} \] Input:
int(x^2/acosh(a*x)^(1/2),x)
Output:
(2*(sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**2 - 3*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**3)/(a**2*x**2 - 1),x)*a**2 + 2*int((s qrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(a**2*x**2 - 1),x)))/a