Integrand size = 28, antiderivative size = 139 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c^3 \sqrt {-1+c x}} \]
1/8*Chi(4*(a+b*arccosh(c*x))/b)*cosh(4*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^( 1/2)-1/8*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-1/8*Shi(4 *(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)
Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {-((-1+c x) (1+c x))} \left (-\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\log (a+b \text {arccosh}(c x))+\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{8 b c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
-1/8*(Sqrt[-((-1 + c*x)*(1 + c*x))]*(-(Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])]) + Log[a + b*ArcCosh[c*x]] + Sinh[(4*a)/b]*SinhIntegral[4* (a/b + ArcCosh[c*x])]))/(b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6367, 5971, 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 |
\(\displaystyle \frac {\sqrt {1-c x} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1}}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\sqrt {1-c x} \int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}-\frac {1}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-c x} \left (\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \log (a+b \text {arccosh}(c x))\right )}{b c^3 \sqrt {c x-1}}\) |
(Sqrt[1 - c*x]*((Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8 - Log[a + b*ArcCosh[c*x]]/8 - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCo sh[c*x]))/b])/8))/(b*c^3*Sqrt[-1 + c*x])
3.3.69.3.1 Defintions of rubi rules used
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
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 = 0.60 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right )}{16 \left (c x +1\right ) c^{3} \left (c x -1\right ) b}\) | \(165\) |
1/16*(-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(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)+Ei(1,-4*arccosh(c*x )-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b))/(c*x+1)/c^3/(c*x-1)/b
\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^2\,\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]