Integrand size = 28, antiderivative size = 139 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c^3 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{2 b c^3 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c^3 \sqrt {1-c x}} \]
1/2*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(c*x-1)^(1/2)/b/c^3/(-c*x+1)^( 1/2)+1/2*ln(a+b*arccosh(c*x))*(c*x-1)^(1/2)/b/c^3/(-c*x+1)^(1/2)-1/2*Shi(2 *(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b/c^3/(-c*x+1)^(1/2)
Time = 0.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\log (a+b \text {arccosh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 c^3 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
-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))
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 |
\(\displaystyle \frac {\sqrt {c x-1} \left (\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{2} \log (a+b \text {arccosh}(c x))\right )}{b c^3 \sqrt {1-c x}}\) |
(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])
3.3.100.3.1 Defintions of rubi rules used
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]))
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]
Time = 0.76 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{4 b \left (c^{2} x^{2}-1\right ) c^{3}}\) | \(161\) |
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
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^{2}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^2}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]