\(\int \frac {1}{(a+b \text {arccosh}(1+d x^2))^{5/2}} \, dx\) [168]

Optimal result

Integrand size = 16, antiderivative size = 252 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=-\frac {2 x^2+d x^4}{3 b x \sqrt {d x^2} \sqrt {2+d x^2} \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{3 b^{5/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{3 b^{5/2} d x} \] Output:

-1/3*(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) 
)^(3/2)-1/3*x/b^2/(a+b*arccosh(d*x^2+1))^(1/2)+1/6*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^(5/2)/d/x+1/6*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^(5/2)/d/x
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {x \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right ) \left (\sqrt {2 \pi } \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {b} \left (-b \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )-\left (a+b \text {arccosh}\left (1+d x^2\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )\right )\right )}{6 b^{5/2} \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2} \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \] Input:

Integrate[(a + b*ArcCosh[1 + d*x^2])^(-5/2),x]
 

Output:

(x*Sinh[ArcCosh[1 + d*x^2]/2]*(Sqrt[2*Pi]*(a + b*ArcCosh[1 + d*x^2])^(3/2) 
*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - S 
inh[a/(2*b)]) + Sqrt[2*Pi]*(a + b*ArcCosh[1 + d*x^2])^(3/2)*Erf[Sqrt[a + b 
*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]) + 
4*Sqrt[b]*(-(b*Cosh[ArcCosh[1 + d*x^2]/2]) - (a + b*ArcCosh[1 + d*x^2])*Si 
nh[ArcCosh[1 + d*x^2]/2])))/(6*b^(5/2)*Sqrt[d*x^2]*Sqrt[(d*x^2)/(2 + d*x^2 
)]*Sqrt[2 + d*x^2]*(a + b*ArcCosh[1 + d*x^2])^(3/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6425, 6419}

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.

\(\displaystyle \int \frac {1}{\left (a+b \text {arccosh}\left (d x^2+1\right )\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6425

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

\(\Big \downarrow \) 6419

\(\displaystyle \frac {\frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}}{3 b^2}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}-\frac {d x^4+2 x^2}{3 b x \sqrt {d x^2} \sqrt {d x^2+2} \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^{3/2}}\)

Input:

Int[(a + b*ArcCosh[1 + d*x^2])^(-5/2),x]
 

Output:

-1/3*(2*x^2 + d*x^4)/(b*x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]*(a + b*ArcCosh[1 + d 
*x^2])^(3/2)) - x/(3*b^2*Sqrt[a + b*ArcCosh[1 + d*x^2]]) + ((Sqrt[Pi/2]*Er 
fi[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])/(Sqrt[b]*d*x) + (Sqrt[Pi/2]*Erf[Sqr 
t[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])/(Sqrt[b]*d*x))/(3*b^2)
 

Defintions of rubi rules used

rule 6419

Int[1/Sqrt[(a_.) + ArcCosh[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[Sqrt 
[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2]*(Erfi[Sqr 
t[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)), 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]]/(Sqrt[b]*d*x)), 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]

\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}+1\right )\right )}^{\frac {5}{2}}}d x\]

Input:

int(1/(a+b*arccosh(d*x^2+1))^(5/2),x)
 

Output:

int(1/(a+b*arccosh(d*x^2+1))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*arccosh(d*x^2+1))^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*acosh(d*x**2+1))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+b*arccosh(d*x^2+1))^(5/2),x, algorithm="maxima")
 

Output:

integrate((b*arccosh(d*x^2 + 1) + a)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+b*arccosh(d*x^2+1))^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x^2 + 1) + a)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^{5/2}} \,d x \] Input:

int(1/(a + b*acosh(d*x^2 + 1))^(5/2),x)
 

Output:

int(1/(a + b*acosh(d*x^2 + 1))^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int(1/(a+b*acosh(d*x^2+1))^(5/2),x)
 

Output:

( - sqrt(d)*acosh(d*x**2 + 1)**2*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)**3*b**3*d*x**2 + 2*acos 
h(d*x**2 + 1)**3*b**3 + 3*acosh(d*x**2 + 1)**2*a*b**2*d*x**2 + 6*acosh(d*x 
**2 + 1)**2*a*b**2 + 3*acosh(d*x**2 + 1)*a**2*b*d*x**2 + 6*acosh(d*x**2 + 
1)*a**2*b + a**3*d*x**2 + 2*a**3),x)*a*b**2*d - sqrt(d)*acosh(d*x**2 + 1)* 
*2*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)**3*b**3*d*x**2 + 2*acosh(d*x**2 + 1)**3*b**3 + 3*a 
cosh(d*x**2 + 1)**2*a*b**2*d*x**2 + 6*acosh(d*x**2 + 1)**2*a*b**2 + 3*acos 
h(d*x**2 + 1)*a**2*b*d*x**2 + 6*acosh(d*x**2 + 1)*a**2*b + a**3*d*x**2 + 2 
*a**3),x)*b**3*d + acosh(d*x**2 + 1)**2*int((sqrt(acosh(d*x**2 + 1)*b + a) 
*acosh(d*x**2 + 1)*x**2)/(acosh(d*x**2 + 1)**3*b**3*d*x**2 + 2*acosh(d*x** 
2 + 1)**3*b**3 + 3*acosh(d*x**2 + 1)**2*a*b**2*d*x**2 + 6*acosh(d*x**2 + 1 
)**2*a*b**2 + 3*acosh(d*x**2 + 1)*a**2*b*d*x**2 + 6*acosh(d*x**2 + 1)*a**2 
*b + a**3*d*x**2 + 2*a**3),x)*b**3*d**2 + 2*acosh(d*x**2 + 1)**2*int((sqrt 
(acosh(d*x**2 + 1)*b + a)*acosh(d*x**2 + 1))/(acosh(d*x**2 + 1)**3*b**3*d* 
x**2 + 2*acosh(d*x**2 + 1)**3*b**3 + 3*acosh(d*x**2 + 1)**2*a*b**2*d*x**2 
+ 6*acosh(d*x**2 + 1)**2*a*b**2 + 3*acosh(d*x**2 + 1)*a**2*b*d*x**2 + 6*ac 
osh(d*x**2 + 1)*a**2*b + a**3*d*x**2 + 2*a**3),x)*b**3*d + sqrt(d)*sqrt(d* 
x**2 + 2)*sqrt(acosh(d*x**2 + 1)*b + a)*acosh(d*x**2 + 1) - 2*sqrt(d)*acos 
h(d*x**2 + 1)*int((sqrt(d*x**2 + 2)*sqrt(acosh(d*x**2 + 1)*b + a)*acosh...