\(\int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [154]

Optimal result

Integrand size = 16, antiderivative size = 231 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \] Output:

-2*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)+1/4*exp(a/ 
b)*Pi^(1/2)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+1/4*exp(3*a/ 
b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/ 
c^3+1/4*Pi^(1/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3/exp(a/ 
b)+1/4*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))/b^( 
3/2)/c^3/exp(3*a/b)
 

Mathematica [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (-2 e^{\frac {3 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-2 e^{\frac {3 a}{b}} \sinh (3 \text {arccosh}(c x))\right )}{4 b c^3 \sqrt {a+b \text {arccosh}(c x)}} \] Input:

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

Output:

(-2*E^((3*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - E^((4*a)/b)*Sqrt[a/ 
b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[-((a + b*A 
rcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + E^((2*a)/b)*Sqr 
t[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - Sqrt[ 
3]*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]) 
)/b] - 2*E^((3*a)/b)*Sinh[3*ArcCosh[c*x]])/(4*b*c^3*E^((3*a)/b)*Sqrt[a + b 
*ArcCosh[c*x]])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6300, 2009}

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.

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

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

Input:

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

Output:

(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2* 
(-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) - ( 
Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt 
[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*E^( 
a/b)) - (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b 
]])/(8*E^((3*a)/b))))/(b^2*c^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6300

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + Simp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

Maple [F]

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

Input:

int(x^2/(a+b*arccosh(c*x))^(3/2),x)
 

Output:

int(x^2/(a+b*arccosh(c*x))^(3/2),x)
 

Fricas [F(-2)]

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

integrate(x^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**2/(a+b*acosh(c*x))**(3/2),x)
 

Output:

Integral(x**2/(a + b*acosh(c*x))**(3/2), x)
 

Maxima [F]

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

integrate(x^2/(a+b*arccosh(c*x))^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^2/(b*arccosh(c*x) + a)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate(x^2/(b*arccosh(c*x) + a)^(3/2), x)
 

Mupad [F(-1)]

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

int(x^2/(a + b*acosh(c*x))^(3/2),x)
 

Output:

int(x^2/(a + b*acosh(c*x))^(3/2), x)
 

Reduce [F]

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

int(x^2/(a+b*acosh(c*x))^(3/2),x)
 

Output:

int((sqrt(acosh(c*x)*b + a)*x**2)/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + 
 a**2),x)