\(\int (a+b \text {arccosh}(-1+d x^2))^{3/2} \, dx\) [171]

Optimal result

Integrand size = 16, antiderivative size = 239 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \, dx=\frac {3 b \left (2 x^2-d x^4\right ) \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{d x}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{d x} \] Output:

3*b*(-d*x^4+2*x^2)*(a+b*arccosh(d*x^2-1))^(1/2)/x/(d*x^2)^(1/2)/(d*x^2-2)^ 
(1/2)+x*(a+b*arccosh(d*x^2-1))^(3/2)+3/2*b^(3/2)*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))/d/x-3/2*b^(3/2)*2^(1/2)*Pi^(1/2)*cosh(1/2*ar 
ccosh(d*x^2-1))*erf(1/2*(a+b*arccosh(d*x^2-1))^(1/2)*2^(1/2)/b^(1/2))*(cos 
h(1/2*a/b)+sinh(1/2*a/b))/d/x
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.92 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \, dx=\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (3 b^{3/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )-3 b^{3/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )} \left (a \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )+b \text {arccosh}\left (-1+d x^2\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )-3 b \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )\right )}{2 d x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6416, 6420}

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.

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

\(\Big \downarrow \) 6416

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

\(\Big \downarrow \) 6420

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

Input:

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

Output:

(3*b*(2*x^2 - d*x^4)*Sqrt[a + b*ArcCosh[-1 + d*x^2]])/(x*Sqrt[d*x^2]*Sqrt[ 
-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^(3/2) + 3*b^2*((Sqrt[Pi/2]*Co 
sh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sq 
rt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(Sqrt[b]*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)]))/(Sqrt[b]*d*x))
 

Defintions of rubi rules used

rule 6416

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

rule 6420

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((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]

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

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

Output:

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

Maxima [F]

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

integrate((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(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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