Integrand size = 25, antiderivative size = 139 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5906, 3393, 3384, 3379, 3382} \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{2 b c \sqrt {c x-1}} \]
[In]
[Out]
Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 5906
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c \sqrt {-1+c x}} \\ & = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{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} \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\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 {\left (\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x}} \\ & = \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}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {-((-1+c x) (1+c x))} \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\log (a+b \text {arccosh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{4 \left (c x -1\right ) \left (c x +1\right ) c b}\) | \(165\) |
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
[In]
[Out]