Integrand size = 12, antiderivative size = 139 \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=-\frac {3 \sqrt {\text {arccosh}(a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{256 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{32 a^4} \] Output:
-3/32*arccosh(a*x)^(1/2)/a^4+1/4*x^4*arccosh(a*x)^(1/2)-1/256*Pi^(1/2)*erf (2*arccosh(a*x)^(1/2))/a^4-1/64*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arccosh(a*x)^ (1/2))/a^4-1/256*Pi^(1/2)*erfi(2*arccosh(a*x)^(1/2))/a^4-1/64*2^(1/2)*Pi^( 1/2)*erfi(2^(1/2)*arccosh(a*x)^(1/2))/a^4
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\frac {\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-4 \text {arccosh}(a x)\right )+4 \sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arccosh}(a x)\right )+\sqrt {-\text {arccosh}(a x)} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},2 \text {arccosh}(a x)\right )+\Gamma \left (\frac {3}{2},4 \text {arccosh}(a x)\right )\right )}{128 a^4 \sqrt {-\text {arccosh}(a x)}} \] Input:
Integrate[x^3*Sqrt[ArcCosh[a*x]],x]
Output:
(Sqrt[ArcCosh[a*x]]*Gamma[3/2, -4*ArcCosh[a*x]] + 4*Sqrt[2]*Sqrt[ArcCosh[a *x]]*Gamma[3/2, -2*ArcCosh[a*x]] + Sqrt[-ArcCosh[a*x]]*(4*Sqrt[2]*Gamma[3/ 2, 2*ArcCosh[a*x]] + Gamma[3/2, 4*ArcCosh[a*x]]))/(128*a^4*Sqrt[-ArcCosh[a *x]])
Time = 1.22 (sec) , antiderivative size = 132, 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.
\(\displaystyle \int x^3 \sqrt {\text {arccosh}(a x)} \, dx\) |
\(\Big \downarrow \) 6299 |
\(\displaystyle \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {1}{8} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1} \sqrt {\text {arccosh}(a x)}}dx\) |
\(\Big \downarrow \) 6368 |
\(\displaystyle \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {a^4 x^4}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )^4}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a^4}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\int \left (\frac {\cosh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}+\frac {\cosh (4 \text {arccosh}(a x))}{8 \sqrt {\text {arccosh}(a x)}}+\frac {3}{8 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{8 a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 \sqrt {\text {arccosh}(a x)}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arccosh}(a x)}}{8 a^4}\) |
Input:
Int[x^3*Sqrt[ArcCosh[a*x]],x]
Output:
(x^4*Sqrt[ArcCosh[a*x]])/4 - ((3*Sqrt[ArcCosh[a*x]])/4 + (Sqrt[Pi]*Erf[2*S qrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[A rcCosh[a*x]]])/4)/(8*a^4)
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]
Time = 0.43 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}-4 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }-\pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{64 \sqrt {\pi }\, a^{4}}+\frac {64 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{4} x^{4}-64 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+8 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }-\pi \,\operatorname {erf}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\pi \,\operatorname {erfi}\left (2 \sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{256 \sqrt {\pi }\, a^{4}}\) | \(154\) |
Input:
int(x^3*arccosh(a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/64*2^(1/2)*(8*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*a^2*x^2-4*2^(1/2)*arcc osh(a*x)^(1/2)*Pi^(1/2)-Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-Pi*erfi(2^(1/2) *arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4+1/256*(64*arccosh(a*x)^(1/2)*Pi^(1/2)*a ^4*x^4-64*arccosh(a*x)^(1/2)*Pi^(1/2)*a^2*x^2+8*arccosh(a*x)^(1/2)*Pi^(1/2 )-Pi*erf(2*arccosh(a*x)^(1/2))-Pi*erfi(2*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^4
Exception generated. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*arccosh(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int x^{3} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \] Input:
integrate(x**3*acosh(a*x)**(1/2),x)
Output:
Integral(x**3*sqrt(acosh(a*x)), x)
\[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int { x^{3} \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \] Input:
integrate(x^3*arccosh(a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^3*sqrt(arccosh(a*x)), x)
Exception generated. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*arccosh(a*x)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int x^3\,\sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \] Input:
int(x^3*acosh(a*x)^(1/2),x)
Output:
int(x^3*acosh(a*x)^(1/2), x)
\[ \int x^3 \sqrt {\text {arccosh}(a x)} \, dx=\int \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}d x \] Input:
int(x^3*acosh(a*x)^(1/2),x)
Output:
int(sqrt(acosh(a*x))*x**3,x)