\(\int \frac {1}{(a+b \text {arccosh}(-1+d x^2))^3} \, dx\) [162]

Optimal result

Integrand size = 14, antiderivative size = 181 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3} \, dx=\frac {2 x^2-d x^4}{4 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}-\frac {x}{8 b^2 \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )}-\frac {x \text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {d x^2}}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {d x^2}} \] Output:

1/4*(-d*x^4+2*x^2)/b/x/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)/(a+b*arccosh(d*x^2-1) 
)^2-1/8*x/b^2/(a+b*arccosh(d*x^2-1))-1/16*x*Chi(1/2*(a+b*arccosh(d*x^2-1)) 
/b)*sinh(1/2*a/b)*2^(1/2)/b^3/(d*x^2)^(1/2)+1/16*x*cosh(1/2*a/b)*Shi(1/2*( 
a+b*arccosh(d*x^2-1))/b)*2^(1/2)/b^3/(d*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3} \, dx=-\frac {\frac {2 b^2 \sqrt {d x^2} \sqrt {-2+d x^2}}{d \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}+\frac {b x^2}{a+b \text {arccosh}\left (-1+d x^2\right )}+\frac {1}{2} \sqrt {1-\frac {2}{d x^2}} x^2 \text {csch}\left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right ) \sinh \left (\frac {a}{2 b}\right )-\cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )\right )}{8 b^3 x} \] Input:

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

Output:

-1/8*((2*b^2*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(d*(a + b*ArcCosh[-1 + d*x^2])^ 
2) + (b*x^2)/(a + b*ArcCosh[-1 + d*x^2]) + (Sqrt[1 - 2/(d*x^2)]*x^2*Csch[A 
rcCosh[-1 + d*x^2]/2]*(CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]*Sin 
h[a/(2*b)] - Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)] 
))/2)/(b^3*x)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6425, 6418}

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.

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

\(\Big \downarrow \) 6425

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

\(\Big \downarrow \) 6418

\(\displaystyle \frac {\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}}{8 b^2}-\frac {x}{8 b^2 \left (a+b \text {arccosh}\left (d x^2-1\right )\right )}+\frac {2 x^2-d x^4}{4 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2}\)

Input:

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

Output:

(2*x^2 - d*x^4)/(4*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d* 
x^2])^2) - x/(8*b^2*(a + b*ArcCosh[-1 + d*x^2])) + (-((x*CoshIntegral[(a + 
 b*ArcCosh[-1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])) + ( 
x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(Sqrt[2]* 
b*Sqrt[d*x^2]))/(8*b^2)
 

Defintions of rubi rules used

rule 6418

Int[((a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> Simp[(-x) 
*Sinh[a/(2*b)]*(CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* 
Sqrt[d*x^2])), x] + Simp[x*Cosh[a/(2*b)]*(SinhIntegral[(a + b*ArcCosh[-1 + 
d*x^2])/(2*b)]/(Sqrt[2]*b*Sqrt[d*x^2])), x] /; FreeQ[{a, b, d}, x]
 

rule 6425

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

Maple [F]

\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/(b^3*arccosh(d*x^2 - 1)^3 + 3*a*b^2*arccosh(d*x^2 - 1)^2 + 3*a^ 
2*b*arccosh(d*x^2 - 1) + a^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/8*((a*d^5 + 2*b*d^5)*sqrt(d)*x^10 - 2*(3*a*d^4 + 7*b*d^4)*sqrt(d)*x^8 + 
 (11*a*d^3 + 36*b*d^3)*sqrt(d)*x^6 - 2*(a*d^2 + 20*b*d^2)*sqrt(d)*x^4 - 4* 
(3*a*d - 4*b*d)*sqrt(d)*x^2 + ((a*d^4 + 2*b*d^4)*x^7 - (3*a*d^3 + 8*b*d^3) 
*x^5 + 2*(2*a*d^2 + 5*b*d^2)*x^3 - 4*(a*d + b*d)*x)*(d*x^2 - 2)^(3/2) + (3 
*(a*d^4 + 2*b*d^4)*sqrt(d)*x^8 - 6*(2*a*d^3 + 5*b*d^3)*sqrt(d)*x^6 + 2*(8* 
a*d^2 + 25*b*d^2)*sqrt(d)*x^4 - 10*(a*d + 3*b*d)*sqrt(d)*x^2 + 4*(a + b)*s 
qrt(d))*(d*x^2 - 2) + (b*d^(11/2)*x^10 - 6*b*d^(9/2)*x^8 + 11*b*d^(7/2)*x^ 
6 - 2*b*d^(5/2)*x^4 - 12*b*d^(3/2)*x^2 + (b*d^4*x^7 - 3*b*d^3*x^5 + 4*b*d^ 
2*x^3 - 4*b*d*x)*(d*x^2 - 2)^(3/2) + (3*b*d^(9/2)*x^8 - 12*b*d^(7/2)*x^6 + 
 16*b*d^(5/2)*x^4 - 10*b*d^(3/2)*x^2 + 4*b*sqrt(d))*(d*x^2 - 2) + (3*b*d^5 
*x^9 - 15*b*d^4*x^7 + 23*b*d^3*x^5 - 7*b*d^2*x^3 - 6*b*d*x)*sqrt(d*x^2 - 2 
) + 8*b*sqrt(d))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1) + (3*(a*d^5 + 
2*b*d^5)*x^9 - 3*(5*a*d^4 + 12*b*d^4)*x^7 + (23*a*d^3 + 76*b*d^3)*x^5 - (7 
*a*d^2 + 64*b*d^2)*x^3 - 2*(3*a*d - 8*b*d)*x)*sqrt(d*x^2 - 2) + 8*a*sqrt(d 
))/(a^2*b^2*d^(11/2)*x^9 - 6*a^2*b^2*d^(9/2)*x^7 + 12*a^2*b^2*d^(7/2)*x^5 
- 8*a^2*b^2*d^(5/2)*x^3 + (b^4*d^(11/2)*x^9 - 6*b^4*d^(9/2)*x^7 + 12*b^4*d 
^(7/2)*x^5 - 8*b^4*d^(5/2)*x^3 + (b^4*d^4*x^6 - 3*b^4*d^3*x^4 + 3*b^4*d^2* 
x^2 - b^4*d)*(d*x^2 - 2)^(3/2) + 3*(b^4*d^(9/2)*x^7 - 4*b^4*d^(7/2)*x^5 + 
5*b^4*d^(5/2)*x^3 - 2*b^4*d^(3/2)*x)*(d*x^2 - 2) + 3*(b^4*d^5*x^8 - 5*b^4* 
d^4*x^6 + 8*b^4*d^3*x^4 - 4*b^4*d^2*x^2)*sqrt(d*x^2 - 2))*log(d*x^2 + s...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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