\(\int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\) [258]

Optimal result

Integrand size = 28, antiderivative size = 139 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c^3 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b c^3 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c^3 \sqrt {1-c x}} \] Output:

1/2*(c*x-1)^(1/2)*cosh(2*a/b)*Chi(2*(a+b*arccosh(c*x))/b)/b/c^3/(-c*x+1)^( 
1/2)+1/2*(c*x-1)^(1/2)*ln(a+b*arccosh(c*x))/b/c^3/(-c*x+1)^(1/2)-1/2*(c*x- 
1)^(1/2)*sinh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)/b/c^3/(-c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\log (a+b \text {arccosh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 c^3 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \] Input:

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

Output:

-1/2*(Sqrt[1 - c^2*x^2]*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x]) 
] + Log[a + b*ArcCosh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[ 
c*x])]))/(c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6367, 3042, 3793, 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.

\(\displaystyle \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\)

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c x-1} \left (\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{2} \log (a+b \text {arccosh}(c x))\right )}{b c^3 \sqrt {1-c x}}\)

Input:

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

Output:

(Sqrt[-1 + c*x]*((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/ 
2 + Log[a + b*ArcCosh[c*x]]/2 - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcC 
osh[c*x]))/b])/2))/(b*c^3*Sqrt[1 - c*x])
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x 
)^p*(-1 + c*x)^p)]   Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p 
 + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{4 b \left (c^{2} x^{2}-1\right ) c^{3}}\)	\(161\)

Input:

int(x^2/(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/4*(-c^2*x^2+1)^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(2*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))-2*ln(a+b*arccosh(c*x))*c*x+Ei 
(1,2*arccosh(c*x)+2*a/b)*exp((-b*arccosh(c*x)+2*a)/b)+Ei(1,-2*arccosh(c*x) 
-2*a/b)*exp(-(b*arccosh(c*x)+2*a)/b))/b/(c^2*x^2-1)/c^3
 

Fricas [F]

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

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

Output:

integral(-sqrt(-c^2*x^2 + 1)*x^2/(a*c^2*x^2 + (b*c^2*x^2 - b)*arccosh(c*x) 
 - a), x)
 

Sympy [F]

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

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

Output:

Integral(x**2/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))), x)
 

Maxima [F]

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

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

Output:

integrate(x^2/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)
 

Giac [F]

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

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

Output:

integrate(x^2/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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