Integrand size = 25, antiderivative size = 139 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}} \]
1/2*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c/(c*x-1)^(1/ 2)-1/2*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c/(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/(c*x-1)^(1/2)
Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {-((-1+c x) (1+c x))} \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\log (a+b \text {arccosh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
(Sqrt[-((-1 + c*x)*(1 + c*x))]*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCos h[c*x])] - Log[a + b*ArcCosh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + A rcCosh[c*x])]))/(2*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
Time = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6321, 3042, 25, 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.
\(\displaystyle \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx\) |
\(\Big \downarrow \) 6321 |
\(\displaystyle \frac {\sqrt {1-c x} \int \frac {\sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {1-c x} \int -\frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^2}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^2}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\sqrt {1-c x} \int \left (\frac {1}{2 (a+b \text {arccosh}(c x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c \sqrt {c x-1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-c x} \left (\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{2} \log (a+b \text {arccosh}(c x))\right )}{b c \sqrt {c x-1}}\) |
(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*ArcCo sh[c*x]))/b])/2))/(b*c*Sqrt[-1 + c*x])
3.3.71.3.1 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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
Time = 0.66 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{4 \left (c x -1\right ) \left (c x +1\right ) c b}\) | \(165\) |
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+E i(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))/(c*x-1)/(c*x+1)/c/b
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]