\(\int (a+b \text {arccosh}(-1+d x^2))^4 \, dx\) [156]

Optimal result

Integrand size = 14, antiderivative size = 147 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=384 b^4 x+\frac {192 b^3 \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+48 b^2 x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \] Output:

384*b^4*x+192*b^3*(-d*x^4+2*x^2)*(a+b*arccosh(d*x^2-1))/x/(d*x^2)^(1/2)/(d 
*x^2-2)^(1/2)+48*b^2*x*(a+b*arccosh(d*x^2-1))^2+8*b*(-d*x^4+2*x^2)*(a+b*ar 
ccosh(d*x^2-1))^3/x/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)+x*(a+b*arccosh(d*x^2-1)) 
^4
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.80 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4+48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {-2+d x^2}+4 b \left (a^3 d x^2+24 a b^2 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {-2+d x^2}-48 b^3 \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )+6 b^2 \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^3+b^4 d x^2 \text {arccosh}\left (-1+d x^2\right )^4}{d x} \] Input:

Integrate[(a + b*ArcCosh[-1 + d*x^2])^4,x]
 

Output:

((a^4 + 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*a*b*(a^2 + 24*b^2)*Sqrt[d*x^2]*Sqr 
t[-2 + d*x^2] + 4*b*(a^3*d*x^2 + 24*a*b^2*d*x^2 - 6*a^2*b*Sqrt[d*x^2]*Sqrt 
[-2 + d*x^2] - 48*b^3*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCosh[-1 + d*x^2] + 
6*b^2*(a^2*d*x^2 + 8*b^2*d*x^2 - 4*a*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCo 
sh[-1 + d*x^2]^2 + 4*b^3*(a*d*x^2 - 2*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcC 
osh[-1 + d*x^2]^3 + b^4*d*x^2*ArcCosh[-1 + d*x^2]^4)/(d*x)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6416, 6416, 24}

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.

\(\displaystyle \int \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^4 \, dx\)

\(\Big \downarrow \) 6416

\(\displaystyle 48 b^2 \int \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2dx+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^4+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

\(\Big \downarrow \) 6416

\(\displaystyle 48 b^2 \left (8 b^2 \int 1dx+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}\right )+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^4+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

\(\Big \downarrow \) 24

\(\displaystyle 48 b^2 \left (x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+8 b^2 x\right )+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^4+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

Input:

Int[(a + b*ArcCosh[-1 + d*x^2])^4,x]
 

Output:

(8*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^3)/(x*Sqrt[d*x^2]*Sqrt[-2 
 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^4 + 48*b^2*(8*b^2*x + (4*b*(2*x 
^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) 
+ x*(a + b*ArcCosh[-1 + d*x^2])^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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. \(1043\) vs. \(2(139)=278\).

Time = 0.21 (sec) , antiderivative size = 1044, normalized size of antiderivative = 7.10

method	result	size
orering	\(\text {Expression too large to display}\)	\(1044\)

Input:

int((a+b*arccosh(d*x^2-1))^4,x,method=_RETURNVERBOSE)
 

Output:

x*(a+b*arccosh(d*x^2-1))^4+32*(a+b*arccosh(d*x^2-1))^3*b*x/(d*x^2-2)^(1/2) 
/(d*x^2)^(1/2)+1/d*x*(5*d*x^2-4)*(48*b^2*(a+b*arccosh(d*x^2-1))^2*d/(d*x^2 
-2)+8*(a+b*arccosh(d*x^2-1))^3*b*d/(d*x^2-2)^(1/2)/(d*x^2)^(1/2)-8*(a+b*ar 
ccosh(d*x^2-1))^3*b*d^2*x^2/(d*x^2-2)^(3/2)/(d*x^2)^(1/2)-8*(a+b*arccosh(d 
*x^2-1))^3*b*d^2*x^2/(d*x^2-2)^(1/2)/(d*x^2)^(3/2))+(5*d^2*x^4-8*d*x^2-4)/ 
d^2*(192*b^3*(a+b*arccosh(d*x^2-1))*d^2/(d*x^2-2)^(3/2)*x/(d*x^2)^(1/2)-14 
4*b^2*(a+b*arccosh(d*x^2-1))^2*d^2/(d*x^2-2)^2*x-24*(a+b*arccosh(d*x^2-1)) 
^3*b*d^2/(d*x^2-2)^(3/2)/(d*x^2)^(1/2)*x-24*(a+b*arccosh(d*x^2-1))^3*b*d^2 
/(d*x^2-2)^(1/2)/(d*x^2)^(3/2)*x+24*(a+b*arccosh(d*x^2-1))^3*b*d^3*x^3/(d* 
x^2-2)^(5/2)/(d*x^2)^(1/2)+16*(a+b*arccosh(d*x^2-1))^3*b*d^3*x^3/(d*x^2-2) 
^(3/2)/(d*x^2)^(3/2)+24*(a+b*arccosh(d*x^2-1))^3*b*d^3*x^3/(d*x^2-2)^(1/2) 
/(d*x^2)^(5/2))+1/d^2*x*(d*x^2-2)^2*(96*(a+b*arccosh(d*x^2-1))^3*b*d^3/(d* 
x^2-2)^(3/2)/(d*x^2)^(3/2)*x^2+192*b^3*(a+b*arccosh(d*x^2-1))*d^2/(d*x^2-2 
)^(3/2)/(d*x^2)^(1/2)-1152*b^3*(a+b*arccosh(d*x^2-1))*d^3/(d*x^2-2)^(5/2)* 
x^2/(d*x^2)^(1/2)-192*b^3*(a+b*arccosh(d*x^2-1))*d^3/(d*x^2-2)^(3/2)*x^2/( 
d*x^2)^(3/2)-24*(a+b*arccosh(d*x^2-1))^3*b*d^2/(d*x^2-2)^(3/2)/(d*x^2)^(1/ 
2)+144*(a+b*arccosh(d*x^2-1))^3*b*d^3/(d*x^2-2)^(5/2)/(d*x^2)^(1/2)*x^2+38 
4*b^4*d^2/(d*x^2-2)^2-192*b^2*(a+b*arccosh(d*x^2-1))^2*d^2/(d*x^2-2)^2+720 
*b^2*(a+b*arccosh(d*x^2-1))^2*d^3/(d*x^2-2)^3*x^2-120*(a+b*arccosh(d*x^2-1 
))^3*b*d^4*x^4/(d*x^2-2)^(7/2)/(d*x^2)^(1/2)-72*(a+b*arccosh(d*x^2-1))^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (137) = 274\).

Time = 0.12 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.03 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{4} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \, {\left (a b^{3} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} - 6 \, {\left (4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 8 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \] Input:

integrate((a+b*arccosh(d*x^2-1))^4,x, algorithm="fricas")
 

Output:

(b^4*d*x^2*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^4 + (a^4 + 48*a^2*b^2 
+ 384*b^4)*d*x^2 + 4*(a*b^3*d*x^2 - 2*sqrt(d^2*x^4 - 2*d*x^2)*b^4)*log(d*x 
^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^3 - 6*(4*sqrt(d^2*x^4 - 2*d*x^2)*a*b^3 - 
 (a^2*b^2 + 8*b^4)*d*x^2)*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^2 + 4*( 
(a^3*b + 24*a*b^3)*d*x^2 - 6*sqrt(d^2*x^4 - 2*d*x^2)*(a^2*b^2 + 8*b^4))*lo 
g(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1) - 8*sqrt(d^2*x^4 - 2*d*x^2)*(a^3*b 
+ 24*a*b^3))/(d*x)
 

Sympy [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \] Input:

integrate((a+b*acosh(d*x**2-1))**4,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*acosh(d*x**2 - 1))**4, x)
 

Maxima [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{4} \,d x } \] Input:

integrate((a+b*arccosh(d*x^2-1))^4,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccosh(d*x^2-1))^4,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. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^4 \,d x \] Input:

int((a + b*acosh(d*x^2 - 1))^4,x)
 

Output:

int((a + b*acosh(d*x^2 - 1))^4, x)
 

Reduce [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=4 \left (\int \mathit {acosh} \left (d \,x^{2}-1\right )d x \right ) a^{3} b +\left (\int \mathit {acosh} \left (d \,x^{2}-1\right )^{4}d x \right ) b^{4}+4 \left (\int \mathit {acosh} \left (d \,x^{2}-1\right )^{3}d x \right ) a \,b^{3}+6 \left (\int \mathit {acosh} \left (d \,x^{2}-1\right )^{2}d x \right ) a^{2} b^{2}+a^{4} x \] Input:

int((a+b*acosh(d*x^2-1))^4,x)
 

Output:

4*int(acosh(d*x**2 - 1),x)*a**3*b + int(acosh(d*x**2 - 1)**4,x)*b**4 + 4*i 
nt(acosh(d*x**2 - 1)**3,x)*a*b**3 + 6*int(acosh(d*x**2 - 1)**2,x)*a**2*b** 
2 + a**4*x