\(\int \frac {1}{(a+b \text {arccosh}(-1+d x^2))^{3/2}} \, dx\) [174]

Optimal result

Integrand size = 16, antiderivative size = 212 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {d x^2} \sqrt {-2+d x^2}}{b d x \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{b^{3/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{b^{3/2} d x} \] Output:

-(d*x^2)^(1/2)*(d*x^2-2)^(1/2)/b/d/x/(a+b*arccosh(d*x^2-1))^(1/2)+1/2*2^(1 
/2)*Pi^(1/2)*cosh(1/2*arccosh(d*x^2-1))*erfi(1/2*(a+b*arccosh(d*x^2-1))^(1 
/2)*2^(1/2)/b^(1/2))*(cosh(1/2*a/b)-sinh(1/2*a/b))/b^(3/2)/d/x+1/2*2^(1/2) 
*Pi^(1/2)*cosh(1/2*arccosh(d*x^2-1))*erf(1/2*(a+b*arccosh(d*x^2-1))^(1/2)* 
2^(1/2)/b^(1/2))*(cosh(1/2*a/b)+sinh(1/2*a/b))/b^(3/2)/d/x
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}} \, dx=\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )-4 \sqrt {b} \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )}{2 b^{3/2} d x \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}} \] Input:

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

Output:

(Cosh[ArcCosh[-1 + d*x^2]/2]*(Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*E 
rfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Si 
nh[a/(2*b)]) + Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*A 
rcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]) - 4 
*Sqrt[b]*Sinh[ArcCosh[-1 + d*x^2]/2]))/(2*b^(3/2)*d*x*Sqrt[a + b*ArcCosh[- 
1 + d*x^2]])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6422}

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.

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

\(\Big \downarrow \) 6422

\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2} d x}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{b d x \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}\)

Input:

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

Output:

-((Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])) 
+ (Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x 
^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(b^(3/2)*d*x) + ( 
Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]] 
/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(b^(3/2)*d*x)
 

Defintions of rubi rules used

rule 6422

Int[((a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.))^(-3/2), x_Symbol] :> Simp[(- 
Sqrt[d*x^2])*(Sqrt[-2 + d*x^2]/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])), x] 
 + (Simp[Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2 
]/2]*(Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)), x] + S 
imp[Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]* 
(Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)), x]) /; Fre 
eQ[{a, b, d}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*sqrt(d*x**2 - 2)*sqrt(acosh(d*x**2 - 1)*b + a)*acosh(d*x**2 - 1) 
- sqrt(d)*acosh(d*x**2 - 1)*int((sqrt(d*x**2 - 2)*sqrt(acosh(d*x**2 - 1)*b 
 + a)*acosh(d*x**2 - 1)*x)/(acosh(d*x**2 - 1)**2*b**2*d*x**2 - 2*acosh(d*x 
**2 - 1)**2*b**2 + 2*acosh(d*x**2 - 1)*a*b*d*x**2 - 4*acosh(d*x**2 - 1)*a* 
b + a**2*d*x**2 - 2*a**2),x)*a*b*d - sqrt(d)*acosh(d*x**2 - 1)*int((sqrt(d 
*x**2 - 2)*sqrt(acosh(d*x**2 - 1)*b + a)*acosh(d*x**2 - 1)**2*x)/(acosh(d* 
x**2 - 1)**2*b**2*d*x**2 - 2*acosh(d*x**2 - 1)**2*b**2 + 2*acosh(d*x**2 - 
1)*a*b*d*x**2 - 4*acosh(d*x**2 - 1)*a*b + a**2*d*x**2 - 2*a**2),x)*b**2*d 
- acosh(d*x**2 - 1)*int((sqrt(acosh(d*x**2 - 1)*b + a)*acosh(d*x**2 - 1)*x 
**2)/(acosh(d*x**2 - 1)**2*b**2*d*x**2 - 2*acosh(d*x**2 - 1)**2*b**2 + 2*a 
cosh(d*x**2 - 1)*a*b*d*x**2 - 4*acosh(d*x**2 - 1)*a*b + a**2*d*x**2 - 2*a* 
*2),x)*b**2*d**2 + 2*acosh(d*x**2 - 1)*int((sqrt(acosh(d*x**2 - 1)*b + a)* 
acosh(d*x**2 - 1))/(acosh(d*x**2 - 1)**2*b**2*d*x**2 - 2*acosh(d*x**2 - 1) 
**2*b**2 + 2*acosh(d*x**2 - 1)*a*b*d*x**2 - 4*acosh(d*x**2 - 1)*a*b + a**2 
*d*x**2 - 2*a**2),x)*b**2*d - sqrt(d)*int((sqrt(d*x**2 - 2)*sqrt(acosh(d*x 
**2 - 1)*b + a)*acosh(d*x**2 - 1)*x)/(acosh(d*x**2 - 1)**2*b**2*d*x**2 - 2 
*acosh(d*x**2 - 1)**2*b**2 + 2*acosh(d*x**2 - 1)*a*b*d*x**2 - 4*acosh(d*x* 
*2 - 1)*a*b + a**2*d*x**2 - 2*a**2),x)*a**2*d - sqrt(d)*int((sqrt(d*x**2 - 
 2)*sqrt(acosh(d*x**2 - 1)*b + a)*acosh(d*x**2 - 1)**2*x)/(acosh(d*x**2 - 
1)**2*b**2*d*x**2 - 2*acosh(d*x**2 - 1)**2*b**2 + 2*acosh(d*x**2 - 1)*a...