\(\int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx\) [66]

Optimal result

Integrand size = 22, antiderivative size = 231 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx=-\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-3/8*a*x^2*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/4*x*(-a^2*c* 
x^2+c)^(1/2)*arccosh(a*x)+3/8*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^2/a/(a*x-1 
)^(1/2)/(a*x+1)^(1/2)-3/4*a*x^2*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^2/(a*x-1 
)^(1/2)/(a*x+1)^(1/2)+1/2*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^3-1/8*(-a^2* 
c*x^2+c)^(1/2)*arccosh(a*x)^4/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.42 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx=-\frac {\sqrt {-c (-1+a x) (1+a x)} \left (2 \text {arccosh}(a x)^4+\left (3+6 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))-2 \text {arccosh}(a x) \left (3+2 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )}{16 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]
 

Output:

-1/16*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(2*ArcCosh[a*x]^4 + (3 + 6*ArcCosh[ 
a*x]^2)*Cosh[2*ArcCosh[a*x]] - 2*ArcCosh[a*x]*(3 + 2*ArcCosh[a*x]^2)*Sinh[ 
2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6310, 6298, 6308, 6354, 15, 6308}

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.

\(\displaystyle \int \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 6354

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}\right )\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {\text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}-\frac {3 a \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {a x-1} \sqrt {a x+1}}\)

Input:

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]
 

Output:

(x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a* 
x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*Sqrt[c - a^2*c*x^2]*((x^2* 
ArcCosh[a*x]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCos 
h[a*x])/(2*a^2) + ArcCosh[a*x]^2/(4*a^3))))/(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x 
])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

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 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{4}}{8 \sqrt {a x -1}\, \sqrt {a x +1}\, a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a}\)	\(256\)

Input:

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4+1 
/32*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1 
)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3-6*arccosh(a*x)^2+6* 
arccosh(a*x)-3)/(a*x-1)/(a*x+1)/a+1/32*(-c*(a^2*x^2-1))^(1/2)*(-2*a^2*x^2* 
(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a^3*x^3+(a*x-1)^(1/2)*(a*x+1)^(1/2)-2*a*x)*( 
4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)/(a*x-1)/(a*x+1)/a
 

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3, x)
 

Sympy [F]

\[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^3,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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx=\sqrt {c}\, \left (\int \sqrt {-a^{2} x^{2}+1}\, \mathit {acosh} \left (a x \right )^{3}d x \right ) \] Input:

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

Output:

sqrt(c)*int(sqrt( - a**2*x**2 + 1)*acosh(a*x)**3,x)