\(\int (a+b \text {arccosh}(1+d x^2))^{3/2} \, dx\) [164]

Optimal result

Integrand size = 16, antiderivative size = 238 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2} \, dx=-\frac {3 b \left (2 x^2+d x^4\right ) \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{d x} \] 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)*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))/d/x+3/2*b^(3/2)*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))*s 
inh(1/2*arccosh(d*x^2+1))/d/x
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.07 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2} \, dx=\frac {x \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right ) \left (3 b^{3/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+3 b^{3/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \left (-3 b \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+a \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+b \text {arccosh}\left (1+d x^2\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )\right )\right )}{2 \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}} \] Input:

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

Output:

(x*Sinh[ArcCosh[1 + d*x^2]/2]*(3*b^(3/2)*Sqrt[2*Pi]*Erfi[Sqrt[a + b*ArcCos 
h[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]]*(-3*b*Cosh[Ar 
cCosh[1 + d*x^2]/2] + a*Sinh[ArcCosh[1 + d*x^2]/2] + b*ArcCosh[1 + d*x^2]* 
Sinh[ArcCosh[1 + d*x^2]/2])))/(2*Sqrt[d*x^2]*Sqrt[(d*x^2)/(2 + d*x^2)]*Sqr 
t[2 + d*x^2])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 242, 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, 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.

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

\(\Big \downarrow \) 6416

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

\(\Big \downarrow \) 6419

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

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]*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])/(Sqrt[b]*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])/(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 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]
 

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