\(\int \frac {1}{(a+b \text {arccosh}(-1+d x^2))^2} \, dx\) [161]

Optimal result

Integrand size = 14, antiderivative size = 150 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2} \, dx=-\frac {\sqrt {d x^2} \sqrt {-2+d x^2}}{2 b d x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )}+\frac {x \cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2} \, dx=\frac {-\frac {b \sqrt {d x^2} \sqrt {-2+d x^2}}{a+b \text {arccosh}\left (-1+d x^2\right )}+\frac {\sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}\left (-1+d x^2\right )}{2 b}\right )\right )}{\sqrt {1-\frac {2}{d x^2}}}}{2 b^2 d x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6424}

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.

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

\(\Big \downarrow \) 6424

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 6424

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*(d^2*x^4 - 3*d*x^2 + (d^(3/2)*x^3 - 2*sqrt(d)*x)*sqrt(d*x^2 - 2) + 2) 
/(a*b*d^2*x^3 - 2*a*b*d*x + (b^2*d^2*x^3 - 2*b^2*d*x + (b^2*d^(3/2)*x^2 - 
b^2*sqrt(d))*sqrt(d*x^2 - 2))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1) + 
 (a*b*d^(3/2)*x^2 - a*b*sqrt(d))*sqrt(d*x^2 - 2)) + integrate(1/2*(d^3*x^6 
 - 3*d^2*x^4 + (d^2*x^4 - d*x^2 + 2)*(d*x^2 - 2) + (2*d^(5/2)*x^5 - 4*d^(3 
/2)*x^3 + sqrt(d)*x)*sqrt(d*x^2 - 2) + 4)/(a*b*d^3*x^6 - 4*a*b*d^2*x^4 + 4 
*a*b*d*x^2 + (a*b*d^2*x^4 - 2*a*b*d*x^2 + a*b)*(d*x^2 - 2) + (b^2*d^3*x^6 
- 4*b^2*d^2*x^4 + 4*b^2*d*x^2 + (b^2*d^2*x^4 - 2*b^2*d*x^2 + b^2)*(d*x^2 - 
 2) + 2*(b^2*d^(5/2)*x^5 - 3*b^2*d^(3/2)*x^3 + 2*b^2*sqrt(d)*x)*sqrt(d*x^2 
 - 2))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1) + 2*(a*b*d^(5/2)*x^5 - 3 
*a*b*d^(3/2)*x^3 + 2*a*b*sqrt(d)*x)*sqrt(d*x^2 - 2)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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