\(\int (a+b \text {arccosh}(-1+d x^2))^3 \, dx\) [157]

Optimal result

Integrand size = 14, antiderivative size = 110 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=24 a b^2 x-48 b^3 \sqrt {1-\frac {2}{d x^2}} x+24 b^3 x \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \] Output:

24*a*b^2*x-48*b^3*(1-2/d/x^2)^(1/2)*x+24*b^3*x*arccosh(d*x^2-1)+6*b*(-d*x^ 
4+2*x^2)*(a+b*arccosh(d*x^2-1))^2/x/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)+x*(a+b*a 
rccosh(d*x^2-1))^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.55 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {a \left (a^2+24 b^2\right ) d x^2-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2} \sqrt {-2+d x^2}+3 b \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )+3 b^2 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^2+b^3 d x^2 \text {arccosh}\left (-1+d x^2\right )^3}{d x} \] Input:

Integrate[(a + b*ArcCosh[-1 + d*x^2])^3,x]
 

Output:

(a*(a^2 + 24*b^2)*d*x^2 - 6*b*(a^2 + 8*b^2)*Sqrt[d*x^2]*Sqrt[-2 + d*x^2] + 
 3*b*(a^2*d*x^2 + 8*b^2*d*x^2 - 4*a*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCos 
h[-1 + d*x^2] + 3*b^2*(a*d*x^2 - 2*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCosh 
[-1 + d*x^2]^2 + b^3*d*x^2*ArcCosh[-1 + d*x^2]^3)/(d*x)
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6416, 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.

\(\displaystyle \int \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3 \, dx\)

\(\Big \downarrow \) 6416

\(\displaystyle 24 b^2 \int \left (a+b \text {arccosh}\left (d x^2-1\right )\right )dx+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle 24 b^2 \left (a x+b x \text {arccosh}\left (d x^2-1\right )-2 b x \sqrt {1-\frac {2}{d x^2}}\right )+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

Input:

Int[(a + b*ArcCosh[-1 + d*x^2])^3,x]
 

Output:

(6*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^2)/(x*Sqrt[d*x^2]*Sqrt[-2 
 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^3 + 24*b^2*(a*x - 2*b*Sqrt[1 - 
2/(d*x^2)]*x + b*x*ArcCosh[-1 + d*x^2])
 

Defintions of rubi rules used

rule 2009

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

rule 6416

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x* 
(a + b*ArcCosh[c + d*x^2])^n, x] + (-Simp[2*b*n*(2*c*d*x^2 + d^2*x^4)*((a + 
 b*ArcCosh[c + d*x^2])^(n - 1)/(d*x*Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2 
])), x] + Simp[4*b^2*n*(n - 1)   Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x] 
, x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(104)=208\).

Time = 0.14 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.30

method	result	size
orering	\(x {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{3}-\frac {6 \left (d \,x^{2}-4\right ) {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b x}{\sqrt {d \,x^{2}-2}\, \sqrt {d \,x^{2}}}-\frac {2 \left (d \,x^{2}-2\right ) x \left (\frac {24 b^{2} d \left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}{d \,x^{2}-2}+\frac {6 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b d}{\sqrt {d \,x^{2}-2}\, \sqrt {d \,x^{2}}}-\frac {6 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{2} x^{2}}{\left (d \,x^{2}-2\right )^{\frac {3}{2}} \sqrt {d \,x^{2}}}-\frac {6 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{2} x^{2}}{\sqrt {d \,x^{2}-2}\, \left (d \,x^{2}\right )^{\frac {3}{2}}}\right )}{d}-\frac {\left (d \,x^{2}-2\right )^{2} \left (\frac {48 b^{3} d^{2} x}{\left (d \,x^{2}-2\right )^{\frac {3}{2}} \sqrt {d \,x^{2}}}-\frac {72 b^{2} d^{2} \left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right ) x}{\left (d \,x^{2}-2\right )^{2}}-\frac {18 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{2} x}{\left (d \,x^{2}-2\right )^{\frac {3}{2}} \sqrt {d \,x^{2}}}-\frac {18 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{2} x}{\sqrt {d \,x^{2}-2}\, \left (d \,x^{2}\right )^{\frac {3}{2}}}+\frac {18 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{3} x^{3}}{\left (d \,x^{2}-2\right )^{\frac {5}{2}} \sqrt {d \,x^{2}}}+\frac {12 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{3} x^{3}}{\left (d \,x^{2}-2\right )^{\frac {3}{2}} \left (d \,x^{2}\right )^{\frac {3}{2}}}+\frac {18 {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{2} b \,d^{3} x^{3}}{\sqrt {d \,x^{2}-2}\, \left (d \,x^{2}\right )^{\frac {5}{2}}}\right )}{d^{2}}\)	\(473\)

Input:

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

Output:

x*(a+b*arccosh(d*x^2-1))^3-6*(d*x^2-4)*(a+b*arccosh(d*x^2-1))^2*b*x/(d*x^2 
-2)^(1/2)/(d*x^2)^(1/2)-2/d*(d*x^2-2)*x*(24*b^2*d*(a+b*arccosh(d*x^2-1))/( 
d*x^2-2)+6*(a+b*arccosh(d*x^2-1))^2*b*d/(d*x^2-2)^(1/2)/(d*x^2)^(1/2)-6*(a 
+b*arccosh(d*x^2-1))^2*b*d^2*x^2/(d*x^2-2)^(3/2)/(d*x^2)^(1/2)-6*(a+b*arcc 
osh(d*x^2-1))^2*b*d^2*x^2/(d*x^2-2)^(1/2)/(d*x^2)^(3/2))-1/d^2*(d*x^2-2)^2 
*(48*b^3*d^2*x/(d*x^2-2)^(3/2)/(d*x^2)^(1/2)-72*b^2*d^2*(a+b*arccosh(d*x^2 
-1))/(d*x^2-2)^2*x-18*(a+b*arccosh(d*x^2-1))^2*b*d^2/(d*x^2-2)^(3/2)/(d*x^ 
2)^(1/2)*x-18*(a+b*arccosh(d*x^2-1))^2*b*d^2/(d*x^2-2)^(1/2)/(d*x^2)^(3/2) 
*x+18*(a+b*arccosh(d*x^2-1))^2*b*d^3*x^3/(d*x^2-2)^(5/2)/(d*x^2)^(1/2)+12* 
(a+b*arccosh(d*x^2-1))^2*b*d^3*x^3/(d*x^2-2)^(3/2)/(d*x^2)^(3/2)+18*(a+b*a 
rccosh(d*x^2-1))^2*b*d^3*x^3/(d*x^2-2)^(1/2)/(d*x^2)^(5/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (103) = 206\).

Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.91 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {b^{3} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \, {\left (a b^{2} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 3 \, {\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \] Input:

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

Output:

(b^3*d*x^2*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^3 + (a^3 + 24*a*b^2)*d 
*x^2 + 3*(a*b^2*d*x^2 - 2*sqrt(d^2*x^4 - 2*d*x^2)*b^3)*log(d*x^2 + sqrt(d^ 
2*x^4 - 2*d*x^2) - 1)^2 + 3*((a^2*b + 8*b^3)*d*x^2 - 4*sqrt(d^2*x^4 - 2*d* 
x^2)*a*b^2)*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1) - 6*sqrt(d^2*x^4 - 2* 
d*x^2)*(a^2*b + 8*b^3))/(d*x)
 

Sympy [F]

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

integrate((a+b*acosh(d*x**2-1))**3,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*acosh(d*x**2 - 1))**3, x)
 

Maxima [F]

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

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

Output:

3*a*b^2*x*arccosh(d*x^2 - 1)^2 + 12*a*b^2*d*(2*x/d - (d^(3/2)*x^2 - 2*sqrt 
(d))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d*x^2) - 1)/(sqrt(d*x^2 - 2)*d^2)) + 
 3*(x*arccosh(d*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(d*x^2 - 2)*d) 
)*a^2*b + (x*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^3 - integrate(6*(d 
^2*x^4 - 2*d*x^2 + (d^(3/2)*x^3 - sqrt(d)*x)*sqrt(d*x^2 - 2))*log(d*x^2 + 
sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^2/(d^2*x^4 - 3*d*x^2 + (d^(3/2)*x^3 - 2*sqr 
t(d)*x)*sqrt(d*x^2 - 2) + 2), x))*b^3 + a^3*x
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:index.cc index_m operator + Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^3 \,d x \] Input:

int((a + b*acosh(d*x^2 - 1))^3,x)
 

Output:

int((a + b*acosh(d*x^2 - 1))^3, x)
 

Reduce [F]

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

int((a+b*acosh(d*x^2-1))^3,x)
 

Output:

3*int(acosh(d*x**2 - 1),x)*a**2*b + int(acosh(d*x**2 - 1)**3,x)*b**3 + 3*i 
nt(acosh(d*x**2 - 1)**2,x)*a*b**2 + a**3*x