Integrand size = 26, antiderivative size = 92 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}} \]
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Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5952, 3384, 3379, 3382} \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}}-\frac {\sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {1-c x}} \\ & = \frac {\left (\sqrt {-1+c x} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {1-c x}}-\frac {\left (\sqrt {-1+c x} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c^2 \sqrt {1-c x}} \\ & = \frac {\sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{c^2 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]
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Time = 0.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}+\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}\right )}{2 b \left (c^{2} x^{2}-1\right ) c^{2}}\) | \(116\) |
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\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
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\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]
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