\(\int \frac {1}{(a+b \text {arccosh}(1+d x^2))^{3/2}} \, dx\) [167]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{b d x \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x} \] 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)*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))*sinh(1/2*arccosh(d*x^2+1))/b^(3/2)/d/x-1/2*2^(1/2) 
*Pi^(1/2)*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))*sinh(1/2*arccosh(d*x^2+1))/b^(3/2)/d/x
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=-\frac {x \left (4 \sqrt {b} \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{2 b^{3/2} \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2} \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}} \] Input:

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

Output:

-1/2*(x*(4*Sqrt[b]*Cosh[ArcCosh[1 + d*x^2]/2] + Sqrt[2*Pi]*Sqrt[a + b*ArcC 
osh[1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(-C 
osh[a/(2*b)] + Sinh[a/(2*b)]) + Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[1 + d*x^2]]* 
Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sin 
h[a/(2*b)]))*Sinh[ArcCosh[1 + d*x^2]/2])/(b^(3/2)*Sqrt[d*x^2]*Sqrt[(d*x^2) 
/(2 + d*x^2)]*Sqrt[2 + d*x^2]*Sqrt[a + b*ArcCosh[1 + d*x^2]])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 213, 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 = {6421}

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.

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

\(\Big \downarrow \) 6421

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

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

Defintions of rubi rules used

rule 6421

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