Integrand size = 12, antiderivative size = 182 \[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{320 a^5}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{320 a^5} \]
-1/1600*erf(5^(1/2)*arccosh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-1/1600*erfi(5 ^(1/2)*arccosh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-1/192*erf(3^(1/2)*arccosh( a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-1/192*erfi(3^(1/2)*arccosh(a*x)^(1/2))*3^ (1/2)*Pi^(1/2)/a^5-1/32*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^5-1/32*erfi(arc cosh(a*x)^(1/2))*Pi^(1/2)/a^5+1/5*x^5*arccosh(a*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89 \[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\frac {3 \sqrt {5} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-5 \text {arccosh}(a x)\right )+25 \sqrt {3} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-3 \text {arccosh}(a x)\right )+150 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-\text {arccosh}(a x)\right )+150 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},\text {arccosh}(a x)\right )+25 \sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},3 \text {arccosh}(a x)\right )+3 \sqrt {5} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},5 \text {arccosh}(a x)\right )}{2400 a^5 \sqrt {-\text {arccosh}(a x)}} \]
(3*Sqrt[5]*Sqrt[ArcCosh[a*x]]*Gamma[3/2, -5*ArcCosh[a*x]] + 25*Sqrt[3]*Sqr t[ArcCosh[a*x]]*Gamma[3/2, -3*ArcCosh[a*x]] + 150*Sqrt[ArcCosh[a*x]]*Gamma [3/2, -ArcCosh[a*x]] + 150*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, ArcCosh[a*x]] + 25*Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, 3*ArcCosh[a*x]] + 3*Sqrt[5]*Sqrt [-ArcCosh[a*x]]*Gamma[3/2, 5*ArcCosh[a*x]])/(2400*a^5*Sqrt[-ArcCosh[a*x]])
Time = 0.77 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6299, 6368, 3042, 3793, 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.
\(\displaystyle \int x^4 \sqrt {\text {arccosh}(a x)} \, dx\) |
\(\Big \downarrow \) 6299 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {a^5 x^5}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{10 a^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^5}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{10 a^5}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {\int \left (\frac {5 a x}{8 \sqrt {\text {arccosh}(a x)}}+\frac {5 \cosh (3 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}+\frac {\cosh (5 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{10 a^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\text {arccosh}(a x)}-\frac {\frac {5}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {5}{32} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )+\frac {5}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {5}{32} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{10 a^5}\) |
(x^5*Sqrt[ArcCosh[a*x]])/5 - ((5*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/16 + (5 *Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi/5]*Erf[Sqrt[5]* Sqrt[ArcCosh[a*x]]])/32 + (5*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/16 + (5*Sq rt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi/5]*Erfi[Sqrt[5]*S qrt[ArcCosh[a*x]]])/32)/(10*a^5)
3.1.72.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x ], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
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]
\[\int x^{4} \sqrt {\operatorname {arccosh}\left (a x \right )}d x\]
Exception generated. \[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\int x^{4} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]
\[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\int { x^{4} \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]
\[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\int { x^{4} \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]
Timed out. \[ \int x^4 \sqrt {\text {arccosh}(a x)} \, dx=\int x^4\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]