\(\int \sqrt {a^2-x^2} \text {arccosh}(\frac {x}{a})^{3/2} \, dx\) [100]

Optimal result

Integrand size = 24, antiderivative size = 316 \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \] Output:

3/16*a*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2)/(-1+x/a)^(1/2)/(x/a+1)^(1/2)-3/8 
*x^2*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2)/a/(-1+x/a)^(1/2)/(x/a+1)^(1/2)+1/2 
*x*(a^2-x^2)^(1/2)*arccosh(x/a)^(3/2)-1/5*a*(a^2-x^2)^(1/2)*arccosh(x/a)^( 
5/2)/(-1+x/a)^(1/2)/(x/a+1)^(1/2)+3/128*a*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2) 
*erf(2^(1/2)*arccosh(x/a)^(1/2))/(-1+x/a)^(1/2)/(x/a+1)^(1/2)+3/128*a*2^(1 
/2)*Pi^(1/2)*(a^2-x^2)^(1/2)*erfi(2^(1/2)*arccosh(x/a)^(1/2))/(-1+x/a)^(1/ 
2)/(x/a+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46 \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a^2 \sqrt {a^2-x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-8 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \left (16 \text {arccosh}\left (\frac {x}{a}\right )^2+15 \cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )-20 \text {arccosh}\left (\frac {x}{a}\right ) \sinh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )\right )\right )}{640 \sqrt {\frac {-a+x}{a+x}} (a+x)} \] Input:

Integrate[Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2),x]
 

Output:

(a^2*Sqrt[a^2 - x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 15*S 
qrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]] - 8*Sqrt[ArcCosh[x/a]]*(16*ArcC 
osh[x/a]^2 + 15*Cosh[2*ArcCosh[x/a]] - 20*ArcCosh[x/a]*Sinh[2*ArcCosh[x/a] 
])))/(640*Sqrt[(-a + x)/(a + x)]*(a + x))
 

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.69, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6310, 6299, 6308, 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.

\(\displaystyle \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx\)

\(\Big \downarrow \) 6310

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \int x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6299

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 6368

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {\sin \left (i \text {arccosh}\left (\frac {x}{a}\right )+\frac {\pi }{2}\right )^2}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \left (\frac {\cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}+\frac {1}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}\right )d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\)

Input:

Int[Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2),x]
 

Output:

(x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/2 - (a*Sqrt[a^2 - x^2]*ArcCosh[x/a] 
^(5/2))/(5*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*Sqrt[a^2 - x^2]*((x^2*Sqrt[A 
rcCosh[x/a]])/2 - (a^2*(Sqrt[ArcCosh[x/a]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ 
ArcCosh[x/a]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/4))/4))/ 
(4*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

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]))
 

rule 6299

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]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n 
, 0]
 

rule 6368

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]
 

Maple [F]

\[\int \sqrt {a^{2}-x^{2}}\, \operatorname {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}d x\]

Input:

int((a^2-x^2)^(1/2)*arccosh(x/a)^(3/2),x)
 

Output:

int((a^2-x^2)^(1/2)*arccosh(x/a)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a^2-x^2)^(1/2)*arccosh(x/a)^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \] Input:

integrate((a**2-x**2)**(1/2)*acosh(x/a)**(3/2),x)
 

Output:

Integral(sqrt(-(-a + x)*(a + x))*acosh(x/a)**(3/2), x)
 

Maxima [F]

\[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { \sqrt {a^{2} - x^{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \] Input:

integrate((a^2-x^2)^(1/2)*arccosh(x/a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(a^2 - x^2)*arccosh(x/a)^(3/2), x)
 

Giac [F]

\[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { \sqrt {a^{2} - x^{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \] Input:

integrate((a^2-x^2)^(1/2)*arccosh(x/a)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(a^2 - x^2)*arccosh(x/a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int {\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}\,\sqrt {a^2-x^2} \,d x \] Input:

int(acosh(x/a)^(3/2)*(a^2 - x^2)^(1/2),x)
 

Output:

int(acosh(x/a)^(3/2)*(a^2 - x^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int \sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right )d x \] Input:

int((a^2-x^2)^(1/2)*acosh(x/a)^(3/2),x)
 

Output:

int(sqrt(a**2 - x**2)*sqrt(acosh(x/a))*acosh(x/a),x)