\(\int (a+b \text {arccosh}(-1+d x^2))^{5/2} \, dx\) [170]

Optimal result

Integrand size = 16, antiderivative size = 281 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2} \, dx=\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}+\frac {30 b^2 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )} \cosh ^2\left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )}{d x}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{d x}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{d x} \] Output:

5*b*(-d*x^4+2*x^2)*(a+b*arccosh(d*x^2-1))^(3/2)/x/(d*x^2)^(1/2)/(d*x^2-2)^ 
(1/2)+x*(a+b*arccosh(d*x^2-1))^(5/2)+30*b^2*(a+b*arccosh(d*x^2-1))^(1/2)*c 
osh(1/2*arccosh(d*x^2-1))^2/d/x-15/2*b^(5/2)*2^(1/2)*Pi^(1/2)*cosh(1/2*arc 
cosh(d*x^2-1))*erfi(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-15/2*b^(5/2)*2^(1/2)*Pi^(1/2)*cosh(1/2*arcco 
sh(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))/d/x
 

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.99 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2} \, dx=\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (-15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )} \left (\left (a^2+15 b^2\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )+b^2 \text {arccosh}\left (-1+d x^2\right )^2 \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )-5 a b \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )+b \text {arccosh}\left (-1+d x^2\right ) \left (2 a \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )-5 b \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )\right )\right )}{2 d x} \] Input:

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

Output:

(Cosh[ArcCosh[-1 + d*x^2]/2]*(-15*b^(5/2)*Sqrt[2*Pi]*Erfi[Sqrt[a + b*ArcCo 
sh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]) - 15*b^ 
(5/2)*Sqrt[2*Pi]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(C 
osh[a/(2*b)] + Sinh[a/(2*b)]) + 4*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*((a^2 + 
15*b^2)*Cosh[ArcCosh[-1 + d*x^2]/2] + b^2*ArcCosh[-1 + d*x^2]^2*Cosh[ArcCo 
sh[-1 + d*x^2]/2] - 5*a*b*Sinh[ArcCosh[-1 + d*x^2]/2] + b*ArcCosh[-1 + d*x 
^2]*(2*a*Cosh[ArcCosh[-1 + d*x^2]/2] - 5*b*Sinh[ArcCosh[-1 + d*x^2]/2])))) 
/(2*d*x)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 284, 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 = {6416, 6415}

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.

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

\(\Big \downarrow \) 6416

\(\displaystyle 15 b^2 \int \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}dx+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{5/2}+\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

\(\Big \downarrow \) 6415

\(\displaystyle 15 b^2 \left (-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{d x}+\frac {2 \cosh ^2\left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{d x}\right )+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{5/2}+\frac {5 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}}{x \sqrt {d x^2} \sqrt {d x^2-2}}\)

Input:

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

Output:

(5*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^(3/2))/(x*Sqrt[d*x^2]*Sqr 
t[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^(5/2) + 15*b^2*((2*Sqrt[a + 
 b*ArcCosh[-1 + d*x^2]]*Cosh[ArcCosh[-1 + d*x^2]/2]^2)/(d*x) - (Sqrt[b]*Sq 
rt[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)]))/(d*x) - (Sqrt[b]*Sqrt[ 
Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqr 
t[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x))
 

Defintions of rubi rules used

rule 6415

Int[Sqrt[(a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[2*Sqr 
t[a + b*ArcCosh[-1 + d*x^2]]*(Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2/(d*x)), x] 
+ (-Simp[Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[(1/2)*ArcC 
osh[-1 + d*x^2]]*(Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d*x)) 
, x] - Simp[Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[(1/2)*A 
rcCosh[-1 + d*x^2]]*(Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d 
*x)), x]) /; FreeQ[{a, b, d}, x]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((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(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2} \, dx=\left (\int \sqrt {\mathit {acosh} \left (d \,x^{2}-1\right ) b +a}d x \right ) a^{2}+2 \left (\int \sqrt {\mathit {acosh} \left (d \,x^{2}-1\right ) b +a}\, \mathit {acosh} \left (d \,x^{2}-1\right )d x \right ) a b +\left (\int \sqrt {\mathit {acosh} \left (d \,x^{2}-1\right ) b +a}\, \mathit {acosh} \left (d \,x^{2}-1\right )^{2}d x \right ) b^{2} \] Input:

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

Output:

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