Integrand size = 10, antiderivative size = 85 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c} \]
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Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5773, 5819, 3384, 3379, 3382} \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {\sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5773
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {c \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx}{b} \\ & = -\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {\sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {\text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {-\frac {b \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-\text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )}{b^2 c} \]
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Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {-\sqrt {c^{2} x^{2}+1}+c x}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {c x +\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\) | \(118\) |
default | \(\frac {\frac {-\sqrt {c^{2} x^{2}+1}+c x}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {c x +\sqrt {c^{2} x^{2}+1}}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\) | \(118\) |
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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